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Properties

Label 40.120.7.a.1
Level $40$
Index $120$
Genus $7$
Analytic rank $4$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $1600$
Index: $120$ $\PSL_2$-index:$120$
Genus: $7 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $10^{4}\cdot20^{4}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $4$
$\Q$-gonality: $4 \le \gamma \le 6$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20C7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.120.7.1

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&4\\16&27\end{bmatrix}$, $\begin{bmatrix}5&4\\24&15\end{bmatrix}$, $\begin{bmatrix}13&10\\10&3\end{bmatrix}$, $\begin{bmatrix}21&38\\32&9\end{bmatrix}$, $\begin{bmatrix}33&26\\18&37\end{bmatrix}$, $\begin{bmatrix}39&4\\6&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.240.7-40.a.1.1, 40.240.7-40.a.1.2, 40.240.7-40.a.1.3, 40.240.7-40.a.1.4, 40.240.7-40.a.1.5, 40.240.7-40.a.1.6, 40.240.7-40.a.1.7, 40.240.7-40.a.1.8, 40.240.7-40.a.1.9, 40.240.7-40.a.1.10, 40.240.7-40.a.1.11, 40.240.7-40.a.1.12, 40.240.7-40.a.1.13, 40.240.7-40.a.1.14, 40.240.7-40.a.1.15, 40.240.7-40.a.1.16, 120.240.7-40.a.1.1, 120.240.7-40.a.1.2, 120.240.7-40.a.1.3, 120.240.7-40.a.1.4, 120.240.7-40.a.1.5, 120.240.7-40.a.1.6, 120.240.7-40.a.1.7, 120.240.7-40.a.1.8, 120.240.7-40.a.1.9, 120.240.7-40.a.1.10, 120.240.7-40.a.1.11, 120.240.7-40.a.1.12, 120.240.7-40.a.1.13, 120.240.7-40.a.1.14, 120.240.7-40.a.1.15, 120.240.7-40.a.1.16, 280.240.7-40.a.1.1, 280.240.7-40.a.1.2, 280.240.7-40.a.1.3, 280.240.7-40.a.1.4, 280.240.7-40.a.1.5, 280.240.7-40.a.1.6, 280.240.7-40.a.1.7, 280.240.7-40.a.1.8, 280.240.7-40.a.1.9, 280.240.7-40.a.1.10, 280.240.7-40.a.1.11, 280.240.7-40.a.1.12, 280.240.7-40.a.1.13, 280.240.7-40.a.1.14, 280.240.7-40.a.1.15, 280.240.7-40.a.1.16
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $6144$

Jacobian

Conductor: $2^{28}\cdot5^{14}$
Simple: no
Squarefree: no
Decomposition: $1^{7}$
Newforms: 50.2.a.b$^{2}$, 100.2.a.a, 1600.2.a.a, 1600.2.a.c, 1600.2.a.o, 1600.2.a.q

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ x t - x u - x v - y w - y t - y u - 2 y v - z w + z t $
$=$ $x w + x u - 2 y t - y u - 2 y v + z w + z t + z u + 2 z v$
$=$ $2 x^{2} + 2 y^{2} + 2 y z - 2 z^{2} + w t + w v + u v + v^{2}$
$=$ $2 x^{2} - 2 x z - 2 y z - 4 z^{2} - w u + t^{2} - t u + t v - u^{2} - u v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 256 x^{10} - 384 x^{9} y - 176 x^{8} y^{2} - 40 x^{8} z^{2} + 752 x^{7} y^{3} + 456 x^{7} y z^{2} + \cdots + y^{2} z^{8} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :

$\displaystyle X$ $=$ $\displaystyle -2x-y-3z$
$\displaystyle Y$ $=$ $\displaystyle 4x+2y+z$
$\displaystyle Z$ $=$ $\displaystyle x+3y-z$

Equation of the image curve:

$0$ $=$ $ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}^+(5)$ $5$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.a.1 $8$ $10$ $10$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.a.1 $8$ $10$ $10$ $0$ $0$ full Jacobian
10.60.3.a.1 $10$ $2$ $2$ $3$ $0$ $1^{4}$
40.60.3.c.1 $40$ $2$ $2$ $3$ $2$ $1^{4}$
40.60.3.ch.1 $40$ $2$ $2$ $3$ $2$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.240.13.a.1 $40$ $2$ $2$ $13$ $4$ $1^{6}$
40.240.13.c.1 $40$ $2$ $2$ $13$ $7$ $1^{6}$
40.240.13.g.1 $40$ $2$ $2$ $13$ $10$ $1^{6}$
40.240.13.i.1 $40$ $2$ $2$ $13$ $5$ $1^{6}$
40.240.13.y.1 $40$ $2$ $2$ $13$ $6$ $1^{6}$
40.240.13.ba.1 $40$ $2$ $2$ $13$ $5$ $1^{6}$
40.240.13.be.1 $40$ $2$ $2$ $13$ $6$ $1^{6}$
40.240.13.bg.1 $40$ $2$ $2$ $13$ $7$ $1^{6}$
40.240.15.a.1 $40$ $2$ $2$ $15$ $10$ $1^{8}$
40.240.15.c.1 $40$ $2$ $2$ $15$ $4$ $1^{8}$
40.240.15.e.1 $40$ $2$ $2$ $15$ $6$ $1^{8}$
40.240.15.g.1 $40$ $2$ $2$ $15$ $7$ $1^{8}$
40.240.15.bb.1 $40$ $2$ $2$ $15$ $7$ $1^{8}$
40.240.15.bd.1 $40$ $2$ $2$ $15$ $7$ $1^{8}$
40.240.15.bi.1 $40$ $2$ $2$ $15$ $9$ $1^{8}$
40.240.15.bk.1 $40$ $2$ $2$ $15$ $5$ $1^{8}$
40.360.19.z.1 $40$ $3$ $3$ $19$ $9$ $1^{12}$
120.240.13.m.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.o.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.s.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.u.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.dg.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.di.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.dm.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.13.do.1 $120$ $2$ $2$ $13$ $?$ not computed
120.240.15.b.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.d.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.i.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.k.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.br.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.bt.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.ck.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.cm.1 $120$ $2$ $2$ $15$ $?$ not computed
280.240.13.bw.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.by.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.cc.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.ce.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.cu.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.cw.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.da.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.13.dc.1 $280$ $2$ $2$ $13$ $?$ not computed
280.240.15.b.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.d.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.i.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.k.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.br.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.bt.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.by.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.ca.1 $280$ $2$ $2$ $15$ $?$ not computed