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Properties

Label 40.12.0.b.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: $4$
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 $2^{2}\cdot4^{2}$ 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: 4E0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.12.0.4

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&4\\0&13\end{bmatrix}$, $\begin{bmatrix}3&10\\26&29\end{bmatrix}$, $\begin{bmatrix}29&20\\12&31\end{bmatrix}$, $\begin{bmatrix}37&4\\10&13\end{bmatrix}$, $\begin{bmatrix}37&22\\8&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.24.0-40.b.1.1, 40.24.0-40.b.1.2, 40.24.0-40.b.1.3, 40.24.0-40.b.1.4, 40.24.0-40.b.1.5, 40.24.0-40.b.1.6, 40.24.0-40.b.1.7, 40.24.0-40.b.1.8, 120.24.0-40.b.1.1, 120.24.0-40.b.1.2, 120.24.0-40.b.1.3, 120.24.0-40.b.1.4, 120.24.0-40.b.1.5, 120.24.0-40.b.1.6, 120.24.0-40.b.1.7, 120.24.0-40.b.1.8, 280.24.0-40.b.1.1, 280.24.0-40.b.1.2, 280.24.0-40.b.1.3, 280.24.0-40.b.1.4, 280.24.0-40.b.1.5, 280.24.0-40.b.1.6, 280.24.0-40.b.1.7, 280.24.0-40.b.1.8
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
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 317 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^6}{5^2}\cdot\frac{x^{12}(25x^{4}+10x^{2}y^{2}+4y^{4})^{3}}{y^{4}x^{16}(5x^{2}+2y^{2})^{2}}$

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
$X(2)$ $2$ $2$ $2$ $0$ $0$
40.6.0.d.1 $40$ $2$ $2$ $0$ $0$
40.6.0.e.1 $40$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
40.24.0.c.1 $40$ $2$ $2$ $0$
40.24.0.d.1 $40$ $2$ $2$ $0$
40.24.0.e.1 $40$ $2$ $2$ $0$
40.24.0.f.1 $40$ $2$ $2$ $0$
40.60.4.d.1 $40$ $5$ $5$ $4$
40.72.3.d.1 $40$ $6$ $6$ $3$
40.120.7.d.1 $40$ $10$ $10$ $7$
120.24.0.k.1 $120$ $2$ $2$ $0$
120.24.0.m.1 $120$ $2$ $2$ $0$
120.24.0.q.1 $120$ $2$ $2$ $0$
120.24.0.s.1 $120$ $2$ $2$ $0$
120.36.2.b.1 $120$ $3$ $3$ $2$
120.48.1.dh.1 $120$ $4$ $4$ $1$
280.24.0.o.1 $280$ $2$ $2$ $0$
280.24.0.p.1 $280$ $2$ $2$ $0$
280.24.0.r.1 $280$ $2$ $2$ $0$
280.24.0.s.1 $280$ $2$ $2$ $0$
280.96.5.b.1 $280$ $8$ $8$ $5$
280.252.16.b.1 $280$ $21$ $21$ $16$
280.336.21.b.1 $280$ $28$ $28$ $21$