| $\GL_2(\Z/31\Z)$-generators: |
$\begin{bmatrix}7&21\\0&17\end{bmatrix}$, $\begin{bmatrix}23&4\\0&6\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
31.64.2-31.a.1.1, 31.64.2-31.a.1.2, 62.64.2-31.a.1.1, 62.64.2-31.a.1.2, 93.64.2-31.a.1.1, 93.64.2-31.a.1.2, 124.64.2-31.a.1.1, 124.64.2-31.a.1.2, 124.64.2-31.a.1.3, 124.64.2-31.a.1.4, 155.64.2-31.a.1.1, 155.64.2-31.a.1.2, 186.64.2-31.a.1.1, 186.64.2-31.a.1.2, 217.64.2-31.a.1.1, 217.64.2-31.a.1.2, 248.64.2-31.a.1.1, 248.64.2-31.a.1.2, 248.64.2-31.a.1.3, 248.64.2-31.a.1.4, 248.64.2-31.a.1.5, 248.64.2-31.a.1.6, 248.64.2-31.a.1.7, 248.64.2-31.a.1.8, 310.64.2-31.a.1.1, 310.64.2-31.a.1.2 |
| Cyclic 31-isogeny field degree: |
$1$ |
| Cyclic 31-torsion field degree: |
$30$ |
| Full 31-torsion field degree: |
$27900$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y^{2} + 2 x y z - x y w - x z^{2} + 8 x z w + 9 x w^{2} + y^{3} + y^{2} z - y^{2} w + y w^{2} + \cdots + z w^{2} $ |
| $=$ | $18 x y^{2} + 6 x y z - 3 x y w - 3 x z^{2} - 7 x z w - 4 x w^{2} + 3 y^{3} + 3 y^{2} z - 2 y^{2} w + \cdots + w^{3}$ |
| $=$ | $4 x y^{2} - 9 x y z - 11 x y w - 11 x z^{2} - 5 x z w + 6 x w^{2} + y^{3} - 2 y^{2} w - 2 y z^{2} + \cdots + z w^{2}$ |
| $=$ | $10 x y^{2} - 7 x y z - 12 x y w + 19 x z^{2} + 3 x z w - 16 x w^{2} + 2 y^{3} - 2 y^{2} w + y z^{2} + \cdots + w^{3}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - x^{3} y + 3 x^{3} z + 2 x^{2} y^{2} - 7 x^{2} y z + 6 x^{2} z^{2} + 5 x y^{3} - 3 x y^{2} z + \cdots + 3 z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x + 1\right) y $ | $=$ | $ -2x^{5} + x^{4} + 4x^{3} - 3x^{2} - 4x - 1 $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y+z+w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -2y^{3}-5y^{2}z-4y^{2}w-4yz^{2}-5yzw-yw^{2}-2z^{3}-3z^{2}w+w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z+w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2}\cdot\frac{14658125312x^{6}w-176015714432x^{5}w^{2}+736561263808x^{4}w^{3}-1196192995048x^{3}w^{4}+534884388100x^{2}w^{5}+2571520204795xyz^{5}+2614513937546xyz^{4}w-676394162313xyz^{3}w^{2}-551315658652xyz^{2}w^{3}+130449315114xyzw^{4}-31868111260xyw^{5}+8780762518705xz^{6}+18030173230883xz^{5}w+6753603380995xz^{4}w^{2}-4195853761659xz^{3}w^{3}-1387215395386xz^{2}w^{4}+116600269930xzw^{5}-289233423084xw^{6}+1078307938305yz^{6}+2079279794795yz^{5}w+714178818509yz^{4}w^{2}-494817261233yz^{3}w^{3}-147371369680yz^{2}w^{4}+6228973180yzw^{5}-30773567820yw^{6}+866341562926z^{7}+1503105409708z^{6}w+146045872879z^{5}w^{2}-562650517415z^{4}w^{3}-10081964733z^{3}w^{4}+33013239083z^{2}w^{5}-27553235294zw^{6}+6950042454w^{7}}{118210688x^{5}w^{2}+9533120x^{4}w^{3}+930248x^{3}w^{4}+101060x^{2}w^{5}+1453249xyz^{5}+30274266xyz^{4}w+140028513xyz^{3}w^{2}+302601968xyz^{2}w^{3}+322326362xyzw^{4}+132730036xyw^{5}+3154563xz^{6}+118643021xz^{5}w+649537601xz^{4}w^{2}+1665883775xz^{3}w^{3}+2301396482xz^{2}w^{4}+1622239834xzw^{5}+451790652xw^{6}+299891yz^{6}+13680357yz^{5}w+75034207yz^{4}w^{2}+191297253yz^{3}w^{3}+261382048yz^{2}w^{4}+181088040yzw^{5}+49223724yw^{6}+338602z^{7}+11827036z^{6}w+61034933z^{5}w^{2}+145982199z^{4}w^{3}+179745681z^{3}w^{4}+96563489z^{2}w^{5}+2428974zw^{6}-10698754w^{7}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
