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Properties

Label 40.24.1.s.1
Level $40$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $32$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $2^{2}$
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: 8B1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.24.1.79

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}11&4\\10&21\end{bmatrix}$, $\begin{bmatrix}21&2\\21&39\end{bmatrix}$, $\begin{bmatrix}21&16\\36&15\end{bmatrix}$, $\begin{bmatrix}21&36\\34&15\end{bmatrix}$, $\begin{bmatrix}23&14\\23&33\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.48.1-40.s.1.1, 40.48.1-40.s.1.2, 40.48.1-40.s.1.3, 40.48.1-40.s.1.4, 40.48.1-40.s.1.5, 40.48.1-40.s.1.6, 40.48.1-40.s.1.7, 40.48.1-40.s.1.8, 40.48.1-40.s.1.9, 40.48.1-40.s.1.10, 40.48.1-40.s.1.11, 40.48.1-40.s.1.12, 80.48.1-40.s.1.1, 80.48.1-40.s.1.2, 80.48.1-40.s.1.3, 80.48.1-40.s.1.4, 80.48.1-40.s.1.5, 80.48.1-40.s.1.6, 80.48.1-40.s.1.7, 80.48.1-40.s.1.8, 120.48.1-40.s.1.1, 120.48.1-40.s.1.2, 120.48.1-40.s.1.3, 120.48.1-40.s.1.4, 120.48.1-40.s.1.5, 120.48.1-40.s.1.6, 120.48.1-40.s.1.7, 120.48.1-40.s.1.8, 120.48.1-40.s.1.9, 120.48.1-40.s.1.10, 120.48.1-40.s.1.11, 120.48.1-40.s.1.12, 240.48.1-40.s.1.1, 240.48.1-40.s.1.2, 240.48.1-40.s.1.3, 240.48.1-40.s.1.4, 240.48.1-40.s.1.5, 240.48.1-40.s.1.6, 240.48.1-40.s.1.7, 240.48.1-40.s.1.8, 280.48.1-40.s.1.1, 280.48.1-40.s.1.2, 280.48.1-40.s.1.3, 280.48.1-40.s.1.4, 280.48.1-40.s.1.5, 280.48.1-40.s.1.6, 280.48.1-40.s.1.7, 280.48.1-40.s.1.8, 280.48.1-40.s.1.9, 280.48.1-40.s.1.10, 280.48.1-40.s.1.11, 280.48.1-40.s.1.12
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $30720$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ y^{2} + y z - z^{2} - w^{2} $
$=$ $20 x^{2} + y w - 2 z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 5 x^{2} y z + 5 y^{2} z^{2} - 25 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{10}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{1000yz^{5}+575yz^{3}w^{2}+60yzw^{4}-625z^{6}-800z^{4}w^{2}-210z^{2}w^{4}-8w^{6}}{w^{4}(5yz-5z^{2}-w^{2})}$

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
8.12.1.b.1 $8$ $2$ $2$ $1$ $0$ dimension zero
20.12.0.i.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
40.12.0.by.1 $40$ $2$ $2$ $0$ $0$ 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
40.48.1.ga.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.ga.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gb.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gb.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.120.9.bi.1 $40$ $5$ $5$ $9$ $1$ $1^{6}\cdot2$
40.144.9.cg.1 $40$ $6$ $6$ $9$ $1$ $1^{6}\cdot2$
40.240.17.lo.1 $40$ $10$ $10$ $17$ $3$ $1^{12}\cdot2^{2}$
80.48.3.cv.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.cw.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.dd.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.dd.2 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.de.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.de.2 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.df.1 $80$ $2$ $2$ $3$ $?$ not computed
80.48.3.dg.1 $80$ $2$ $2$ $3$ $?$ not computed
120.48.1.rk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.rk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.rl.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.rl.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.5.co.1 $120$ $3$ $3$ $5$ $?$ not computed
120.96.5.bs.1 $120$ $4$ $4$ $5$ $?$ not computed
240.48.3.dl.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.dm.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.dt.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.dt.2 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.du.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.du.2 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.dv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.48.3.dw.1 $240$ $2$ $2$ $3$ $?$ not computed
280.48.1.oa.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.oa.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ob.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.ob.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.192.13.bs.1 $280$ $8$ $8$ $13$ $?$ not computed