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Properties

Label 40.12.0.ba.1
Level $40$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $8$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $1^{2}\cdot2\cdot8$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8C0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.12.0.9

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&4\\10&9\end{bmatrix}$, $\begin{bmatrix}9&32\\12&15\end{bmatrix}$, $\begin{bmatrix}23&24\\30&33\end{bmatrix}$, $\begin{bmatrix}25&12\\13&9\end{bmatrix}$, $\begin{bmatrix}27&8\\29&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.24.0-40.ba.1.1, 40.24.0-40.ba.1.2, 40.24.0-40.ba.1.3, 40.24.0-40.ba.1.4, 40.24.0-40.ba.1.5, 40.24.0-40.ba.1.6, 40.24.0-40.ba.1.7, 40.24.0-40.ba.1.8, 40.24.0-40.ba.1.9, 40.24.0-40.ba.1.10, 40.24.0-40.ba.1.11, 40.24.0-40.ba.1.12, 40.24.0-40.ba.1.13, 40.24.0-40.ba.1.14, 40.24.0-40.ba.1.15, 40.24.0-40.ba.1.16, 120.24.0-40.ba.1.1, 120.24.0-40.ba.1.2, 120.24.0-40.ba.1.3, 120.24.0-40.ba.1.4, 120.24.0-40.ba.1.5, 120.24.0-40.ba.1.6, 120.24.0-40.ba.1.7, 120.24.0-40.ba.1.8, 120.24.0-40.ba.1.9, 120.24.0-40.ba.1.10, 120.24.0-40.ba.1.11, 120.24.0-40.ba.1.12, 120.24.0-40.ba.1.13, 120.24.0-40.ba.1.14, 120.24.0-40.ba.1.15, 120.24.0-40.ba.1.16, 280.24.0-40.ba.1.1, 280.24.0-40.ba.1.2, 280.24.0-40.ba.1.3, 280.24.0-40.ba.1.4, 280.24.0-40.ba.1.5, 280.24.0-40.ba.1.6, 280.24.0-40.ba.1.7, 280.24.0-40.ba.1.8, 280.24.0-40.ba.1.9, 280.24.0-40.ba.1.10, 280.24.0-40.ba.1.11, 280.24.0-40.ba.1.12, 280.24.0-40.ba.1.13, 280.24.0-40.ba.1.14, 280.24.0-40.ba.1.15, 280.24.0-40.ba.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $61440$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 960 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 12 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5^4}\cdot\frac{x^{12}(25x^{4}+80x^{2}y^{2}+16y^{4})^{3}}{y^{2}x^{20}(5x^{2}+y^{2})}$

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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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(4)$ $4$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
40.24.0.m.1 $40$ $2$ $2$ $0$
40.24.0.n.1 $40$ $2$ $2$ $0$
40.24.0.be.1 $40$ $2$ $2$ $0$
40.24.0.bg.1 $40$ $2$ $2$ $0$
40.24.0.bj.1 $40$ $2$ $2$ $0$
40.24.0.bk.1 $40$ $2$ $2$ $0$
40.24.0.bw.1 $40$ $2$ $2$ $0$
40.24.0.bz.1 $40$ $2$ $2$ $0$
40.60.4.bo.1 $40$ $5$ $5$ $4$
40.72.3.ce.1 $40$ $6$ $6$ $3$
40.120.7.cu.1 $40$ $10$ $10$ $7$
120.24.0.by.1 $120$ $2$ $2$ $0$
120.24.0.ca.1 $120$ $2$ $2$ $0$
120.24.0.cg.1 $120$ $2$ $2$ $0$
120.24.0.ci.1 $120$ $2$ $2$ $0$
120.24.0.dp.1 $120$ $2$ $2$ $0$
120.24.0.dq.1 $120$ $2$ $2$ $0$
120.24.0.ea.1 $120$ $2$ $2$ $0$
120.24.0.ed.1 $120$ $2$ $2$ $0$
120.36.2.di.1 $120$ $3$ $3$ $2$
120.48.1.baa.1 $120$ $4$ $4$ $1$
280.24.0.cu.1 $280$ $2$ $2$ $0$
280.24.0.cw.1 $280$ $2$ $2$ $0$
280.24.0.cy.1 $280$ $2$ $2$ $0$
280.24.0.da.1 $280$ $2$ $2$ $0$
280.24.0.ds.1 $280$ $2$ $2$ $0$
280.24.0.du.1 $280$ $2$ $2$ $0$
280.24.0.ea.1 $280$ $2$ $2$ $0$
280.24.0.ec.1 $280$ $2$ $2$ $0$
280.96.5.cc.1 $280$ $8$ $8$ $5$
280.252.16.di.1 $280$ $21$ $21$ $16$
280.336.21.di.1 $280$ $28$ $28$ $21$