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Properties

Label 40.72.1.bh.2
Level $40$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $400$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $2^{6}\cdot10^{6}$ Cusp orbits $2^{2}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.197

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}15&22\\38&39\end{bmatrix}$, $\begin{bmatrix}25&4\\34&25\end{bmatrix}$, $\begin{bmatrix}29&0\\36&23\end{bmatrix}$, $\begin{bmatrix}31&17\\30&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.144.1-40.bh.2.1, 80.144.1-40.bh.2.2, 80.144.1-40.bh.2.3, 80.144.1-40.bh.2.4, 80.144.1-40.bh.2.5, 80.144.1-40.bh.2.6, 80.144.1-40.bh.2.7, 80.144.1-40.bh.2.8, 80.144.1-40.bh.2.9, 80.144.1-40.bh.2.10, 80.144.1-40.bh.2.11, 80.144.1-40.bh.2.12, 80.144.1-40.bh.2.13, 80.144.1-40.bh.2.14, 80.144.1-40.bh.2.15, 80.144.1-40.bh.2.16, 240.144.1-40.bh.2.1, 240.144.1-40.bh.2.2, 240.144.1-40.bh.2.3, 240.144.1-40.bh.2.4, 240.144.1-40.bh.2.5, 240.144.1-40.bh.2.6, 240.144.1-40.bh.2.7, 240.144.1-40.bh.2.8, 240.144.1-40.bh.2.9, 240.144.1-40.bh.2.10, 240.144.1-40.bh.2.11, 240.144.1-40.bh.2.12, 240.144.1-40.bh.2.13, 240.144.1-40.bh.2.14, 240.144.1-40.bh.2.15, 240.144.1-40.bh.2.16
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
Full 40-torsion field degree: $10240$

Jacobian

Conductor: $2^{4}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 400.2.a.c

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x^{2} - x w + 2 y^{2} $
$=$ $2 x^{2} - 6 x w - 4 y^{2} - 10 z^{2} + 5 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 10 x^{2} y^{2} + 4 x^{2} z^{2} + 20 z^{4} $
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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 embedded model to plane model:

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

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{76800000000xz^{16}w-51200000000xz^{14}w^{3}+14144000000xz^{12}w^{5}-2073600000xz^{10}w^{7}+169600000xz^{8}w^{9}-7072000xz^{6}w^{11}+72000xz^{4}w^{13}+4320xz^{2}w^{15}-108xw^{17}+64000000000z^{18}-57600000000z^{16}w^{2}+12800000000z^{14}w^{4}+1648000000z^{12}w^{6}-1200000000z^{10}w^{8}+227840000z^{8}w^{10}-22196000z^{6}w^{12}+1206000z^{4}w^{14}-34560z^{2}w^{16}+405w^{18}}{w^{10}(10z^{2}-w^{2})^{2}(80xz^{2}w-4xw^{3}+400z^{4}-230z^{2}w^{2}+15w^{4})}$

Modular covers

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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
20.36.1.d.1 $20$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.cr.1 $40$ $3$ $3$ $1$ $0$ dimension zero
40.36.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.f.2 $40$ $2$ $2$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.144.5.fd.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.fg.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.gf.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.gi.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.in.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.ir.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.jp.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.jt.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.360.13.cs.1 $40$ $5$ $5$ $13$ $1$ $1^{6}\cdot2^{3}$
120.144.5.cnh.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cnk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cpl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cpo.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eef.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eej.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.uf.1 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.iej.1 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bh.2 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bex.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bey.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfl.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfm.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bnn.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bnp.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bob.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bod.2 $280$ $2$ $2$ $5$ $?$ not computed