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Properties

Label 40.24.1.c.1
Level $40$
Index $24$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $1^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8B1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.24.1.2

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&28\\10&29\end{bmatrix}$, $\begin{bmatrix}9&8\\18&15\end{bmatrix}$, $\begin{bmatrix}11&12\\12&5\end{bmatrix}$, $\begin{bmatrix}21&4\\12&7\end{bmatrix}$, $\begin{bmatrix}21&32\\2&3\end{bmatrix}$, $\begin{bmatrix}29&24\\10&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.48.1-40.c.1.1, 40.48.1-40.c.1.2, 40.48.1-40.c.1.3, 40.48.1-40.c.1.4, 40.48.1-40.c.1.5, 40.48.1-40.c.1.6, 40.48.1-40.c.1.7, 40.48.1-40.c.1.8, 40.48.1-40.c.1.9, 40.48.1-40.c.1.10, 40.48.1-40.c.1.11, 40.48.1-40.c.1.12, 40.48.1-40.c.1.13, 40.48.1-40.c.1.14, 40.48.1-40.c.1.15, 40.48.1-40.c.1.16, 40.48.1-40.c.1.17, 40.48.1-40.c.1.18, 40.48.1-40.c.1.19, 40.48.1-40.c.1.20, 120.48.1-40.c.1.1, 120.48.1-40.c.1.2, 120.48.1-40.c.1.3, 120.48.1-40.c.1.4, 120.48.1-40.c.1.5, 120.48.1-40.c.1.6, 120.48.1-40.c.1.7, 120.48.1-40.c.1.8, 120.48.1-40.c.1.9, 120.48.1-40.c.1.10, 120.48.1-40.c.1.11, 120.48.1-40.c.1.12, 120.48.1-40.c.1.13, 120.48.1-40.c.1.14, 120.48.1-40.c.1.15, 120.48.1-40.c.1.16, 120.48.1-40.c.1.17, 120.48.1-40.c.1.18, 120.48.1-40.c.1.19, 120.48.1-40.c.1.20, 280.48.1-40.c.1.1, 280.48.1-40.c.1.2, 280.48.1-40.c.1.3, 280.48.1-40.c.1.4, 280.48.1-40.c.1.5, 280.48.1-40.c.1.6, 280.48.1-40.c.1.7, 280.48.1-40.c.1.8, 280.48.1-40.c.1.9, 280.48.1-40.c.1.10, 280.48.1-40.c.1.11, 280.48.1-40.c.1.12, 280.48.1-40.c.1.13, 280.48.1-40.c.1.14, 280.48.1-40.c.1.15, 280.48.1-40.c.1.16, 280.48.1-40.c.1.17, 280.48.1-40.c.1.18, 280.48.1-40.c.1.19, 280.48.1-40.c.1.20
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $30720$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 25x $
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Rational points

This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{5^2}\cdot\frac{451875x^{2}y^{4}z^{2}+39990234375x^{2}z^{6}+1150xy^{6}z+3200390625xy^{2}z^{5}+y^{8}+64687500y^{4}z^{4}+244140625z^{8}}{zy^{4}(25x^{2}z-xy^{2}-625z^{3})}$

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
$X_{\pm1}(2,4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
40.12.0.by.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.12.1.f.1 $40$ $2$ $2$ $1$ $1$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.48.1.o.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.v.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bc.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bc.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bd.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bd.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.be.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.be.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bf.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bf.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bt.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.48.1.bu.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.120.9.g.1 $40$ $5$ $5$ $9$ $2$ $1^{6}\cdot2$
40.144.9.g.1 $40$ $6$ $6$ $9$ $3$ $1^{6}\cdot2$
40.240.17.cg.1 $40$ $10$ $10$ $17$ $3$ $1^{12}\cdot2^{2}$
120.48.1.bs.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dl.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dl.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dm.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dm.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.dn.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.ez.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.fd.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.5.c.1 $120$ $3$ $3$ $5$ $?$ not computed
120.96.5.c.1 $120$ $4$ $4$ $5$ $?$ not computed
280.48.1.bt.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.bv.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ca.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ca.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cb.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cb.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cc.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cc.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cd.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cd.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.cz.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.db.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.192.13.c.1 $280$ $8$ $8$ $13$ $?$ not computed