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WeierstrassInvariantG3
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WeierstrassInvariantG3

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gives the invariant for the Weierstrass elliptic functions corresponding to the half‐periods {ω,ω′}.

Details

Examples

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Basic Examples  (2)Summary of the most common use cases

Evaluate numerically:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-e1cujh
Direct link to example
Out[1]=1

Plot the invariant:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-i4z09g
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Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Evaluate to high precision:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-6pbu
Direct link to example
Out[1]=1

The precision of the output tracks the precision of the input:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-idoarf
Direct link to example
Out[1]=1

Evaluate symbolically for the equianharmonic case:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-dmeqrb
Direct link to example
Out[1]=1

Evaluate symbolically for the lemniscatic case:

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In[2]:=2
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https://wolfram.com/xid/0rkvmtmat9un5i9-bs95ea
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Out[2]=2

WeierstrassInvariantG3 has both singularities and discontinuities:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-mdtl3h
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0rkvmtmat9un5i9-mn5jws
Direct link to example
Out[2]=2

WeierstrassInvariantG3 can be used with CenteredInterval objects:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-pgrx3
Direct link to example
Out[1]=1

TraditionalForm formatting:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-ej96jw
Direct link to example

Applications  (1)Sample problems that can be solved with this function

Define the discriminant of the Weierstrass elliptic curve:

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In[1]:=1
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https://wolfram.com/xid/0rkvmtmat9un5i9-bxz1na
Direct link to example
Out[1]=1

KleinInvariantJ can be computed as the ratio of a power of invariant and the discriminant:

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In[2]:=2
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https://wolfram.com/xid/0rkvmtmat9un5i9-m7mjw
Direct link to example
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In[3]:=3
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https://wolfram.com/xid/0rkvmtmat9un5i9-mjpanl
Direct link to example
Out[3]=3

Compare with the built‐in function value:

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In[4]:=4
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https://wolfram.com/xid/0rkvmtmat9un5i9-cm8vjo
Direct link to example
Out[4]=4
Wolfram Research (2017), WeierstrassInvariantG3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html (updated 2023).
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Wolfram Research (2017), WeierstrassInvariantG3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html (updated 2023).

Text

Wolfram Research (2017), WeierstrassInvariantG3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html (updated 2023).

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Wolfram Research (2017), WeierstrassInvariantG3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html (updated 2023).

CMS

Wolfram Language. 2017. "WeierstrassInvariantG3." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html.

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Wolfram Language. 2017. "WeierstrassInvariantG3." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html.

APA

Wolfram Language. (2017). WeierstrassInvariantG3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html

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Wolfram Language. (2017). WeierstrassInvariantG3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html

BibTeX

@misc{reference.wolfram_2025_weierstrassinvariantg3, author="Wolfram Research", title="{WeierstrassInvariantG3}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_weierstrassinvariantg3, author="Wolfram Research", title="{WeierstrassInvariantG3}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weierstrassinvariantg3, organization={Wolfram Research}, title={WeierstrassInvariantG3}, year={2023}, url={https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_weierstrassinvariantg3, organization={Wolfram Research}, title={WeierstrassInvariantG3}, year={2023}, url={https://reference.wolfram.com/language/ref/WeierstrassInvariantG3.html}, note=[Accessed: 26-March-2025 ]}