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Properties

Label 40.120.5.cu.1
Level $40$
Index $120$
Genus $5$
Analytic rank $2$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $1600$
Index: $120$ $\PSL_2$-index:$120$
Genus: $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $10^{12}$ Cusp orbits $1^{2}\cdot2\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10A5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.120.5.4

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&35\\5&6\end{bmatrix}$, $\begin{bmatrix}13&21\\20&39\end{bmatrix}$, $\begin{bmatrix}17&2\\10&9\end{bmatrix}$, $\begin{bmatrix}31&38\\0&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.240.5-40.cu.1.1, 40.240.5-40.cu.1.2, 40.240.5-40.cu.1.3, 40.240.5-40.cu.1.4, 40.240.5-40.cu.1.5, 40.240.5-40.cu.1.6, 40.240.5-40.cu.1.7, 40.240.5-40.cu.1.8, 120.240.5-40.cu.1.1, 120.240.5-40.cu.1.2, 120.240.5-40.cu.1.3, 120.240.5-40.cu.1.4, 120.240.5-40.cu.1.5, 120.240.5-40.cu.1.6, 120.240.5-40.cu.1.7, 120.240.5-40.cu.1.8, 280.240.5-40.cu.1.1, 280.240.5-40.cu.1.2, 280.240.5-40.cu.1.3, 280.240.5-40.cu.1.4, 280.240.5-40.cu.1.5, 280.240.5-40.cu.1.6, 280.240.5-40.cu.1.7, 280.240.5-40.cu.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $6144$

Jacobian

Conductor: $2^{30}\cdot5^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 320.2.a.f, 1600.2.a.c$^{2}$, 1600.2.c.d

Models

Embedded model Embedded model in $\mathbb{P}^{6}$

$ 0 $ $=$ $ z^{2} v - z w v + z u v + w t v + t u v + u^{2} v $
$=$ $z^{2} u - z w u + z u^{2} + w t u + t u^{2} + u^{3}$
$=$ $z^{3} - z^{2} w + z^{2} u + z w t + z t u + z u^{2}$
$=$ $z^{2} t - z w t + z t u + w t^{2} + t^{2} u + t u^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 11 x^{7} + 53 x^{6} z + 4 x^{5} y^{2} + 81 x^{5} z^{2} + 30 x^{4} y^{2} z + 40 x^{4} z^{3} + \cdots - 11 z^{7} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -10x^{11} - 110x^{6} + 10x $
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Rational points

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.

Embedded model
$(0:0:0:0:0:0:1)$, $(-1:-1:-1:-1:-1:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}v$
$\displaystyle Z$ $=$ $\displaystyle y$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -\frac{2}{5}x-\frac{3}{5}y$
$\displaystyle Y$ $=$ $\displaystyle -\frac{2}{625}x^{5}v-\frac{3}{125}x^{4}yv-\frac{14}{125}x^{3}y^{2}v-\frac{36}{125}x^{2}y^{3}v-\frac{47}{125}xy^{4}v-\frac{123}{625}y^{5}v$
$\displaystyle Z$ $=$ $\displaystyle -\frac{1}{5}x+\frac{1}{5}y$

Maps to other modular curves

$j$-invariant map of degree 120 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{558149400000xy^{10}-699522450000xy^{8}v^{2}+1461126510000xy^{6}v^{4}-3990432286500xy^{4}v^{6}+12316523419350xy^{2}v^{8}+1112359519506300000xu^{10}+546969634677930000xu^{8}v^{2}-108060717349269000xu^{6}v^{4}+3151734834390900xu^{4}v^{6}+789616896353400xu^{2}v^{8}-40842196858737xv^{10}+213194100000y^{11}-1651313550000y^{9}v^{2}+3827814915000y^{7}v^{4}-10476089076000y^{5}v^{6}+32325266464650y^{3}v^{8}+64614113317200000yu^{10}+301379392919520000yu^{8}v^{2}-38824583451546000yu^{6}v^{4}+1846283646738600yu^{4}v^{6}+331777489856100yu^{2}v^{8}-107173872075168yv^{10}+235394880044400000ztu^{9}+152592035376510000ztu^{7}v^{2}-18313915447974000ztu^{5}v^{4}+629191214755500ztu^{3}v^{6}+301477907455500ztuv^{8}-932436140350000000zu^{10}-1017832128660430000zu^{8}v^{2}+61905306433378000zu^{6}v^{4}+2887535523917300zu^{4}v^{6}-1440131422253700zu^{2}v^{8}-50489059569750zv^{10}-697041259518100000wtu^{9}-946669141734070000wtu^{7}v^{2}+136236512535514000wtu^{5}v^{4}-3523168518617200wtu^{3}v^{6}-941293224597450wtuv^{8}+219048544660100000wu^{10}+93568054674650000wu^{8}v^{2}-44702596253252000wu^{6}v^{4}+550580222719400wu^{4}v^{6}-46534006105350wu^{2}v^{8}+6563271950625wv^{10}+454443426279500000t^{2}u^{9}+177896571072410000t^{2}u^{7}v^{2}-41481218118626000t^{2}u^{5}v^{4}+1420907702138900t^{2}u^{3}v^{6}+237247384000650t^{2}uv^{8}-235394879556900000tu^{10}-35691997375260000tu^{8}v^{2}+16814304448584000tu^{6}v^{4}-5545762338177000tu^{4}v^{6}-180542073884250tu^{2}v^{8}+18252416536125tv^{10}+235394882106900000u^{11}+195564484337610000u^{9}v^{2}+58305179839446000u^{7}v^{4}-9634000334916000u^{5}v^{6}+571878966168600u^{3}v^{8}+78143326140225uv^{10}}{v^{10}(2x+3y+4z+8w-3t-11u)}$

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
5.60.0.a.1 $5$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.cp.1 $40$ $5$ $5$ $1$ $1$ $1^{2}\cdot2$
40.24.1.cp.2 $40$ $5$ $5$ $1$ $1$ $1^{2}\cdot2$
40.60.2.e.1 $40$ $2$ $2$ $2$ $0$ $1^{3}$
40.60.3.bj.1 $40$ $2$ $2$ $3$ $2$ $2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.360.13.ci.1 $40$ $3$ $3$ $13$ $3$ $1^{4}\cdot2^{2}$
40.480.29.bbo.1 $40$ $4$ $4$ $29$ $6$ $1^{12}\cdot2^{6}$