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Properties

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

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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: $4$ (none of which are rational) Cusp widths $2^{2}\cdot4^{2}$ Cusp orbits $2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1 \le \gamma \le 2$
$\overline{\Q}$-gonality: $1$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 4E0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.12.0.19

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&8\\14&31\end{bmatrix}$, $\begin{bmatrix}5&26\\3&3\end{bmatrix}$, $\begin{bmatrix}13&2\\11&31\end{bmatrix}$, $\begin{bmatrix}13&18\\9&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.24.0-40.bh.1.1, 80.24.0-40.bh.1.2, 80.24.0-40.bh.1.3, 80.24.0-40.bh.1.4, 80.24.0-40.bh.1.5, 80.24.0-40.bh.1.6, 80.24.0-40.bh.1.7, 80.24.0-40.bh.1.8, 240.24.0-40.bh.1.1, 240.24.0-40.bh.1.2, 240.24.0-40.bh.1.3, 240.24.0-40.bh.1.4, 240.24.0-40.bh.1.5, 240.24.0-40.bh.1.6, 240.24.0-40.bh.1.7, 240.24.0-40.bh.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 6 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^6}{3^4}\cdot\frac{(5x+y)^{12}(325x^{4}+800x^{3}y+1500x^{2}y^{2}+320xy^{3}+52y^{4})^{3}}{(5x+y)^{12}(5x^{2}-2y^{2})^{4}(5x^{2}+20xy+2y^{2})^{2}}$

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
4.6.0.d.1 $4$ $2$ $2$ $0$ $0$
40.6.0.b.1 $40$ $2$ $2$ $0$ $0$
40.6.0.c.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
40.24.1.v.1 $40$ $2$ $2$ $1$
40.24.1.x.1 $40$ $2$ $2$ $1$
40.24.1.ef.1 $40$ $2$ $2$ $1$
40.24.1.eh.1 $40$ $2$ $2$ $1$
40.60.4.bx.1 $40$ $5$ $5$ $4$
40.72.3.dx.1 $40$ $6$ $6$ $3$
40.120.7.fx.1 $40$ $10$ $10$ $7$
120.24.1.eb.1 $120$ $2$ $2$ $1$
120.24.1.ed.1 $120$ $2$ $2$ $1$
120.24.1.pd.1 $120$ $2$ $2$ $1$
120.24.1.pf.1 $120$ $2$ $2$ $1$
120.36.2.qp.1 $120$ $3$ $3$ $2$
120.48.1.bzn.1 $120$ $4$ $4$ $1$
280.24.1.da.1 $280$ $2$ $2$ $1$
280.24.1.db.1 $280$ $2$ $2$ $1$
280.24.1.fm.1 $280$ $2$ $2$ $1$
280.24.1.fn.1 $280$ $2$ $2$ $1$
280.96.5.fd.1 $280$ $8$ $8$ $5$
280.252.16.kb.1 $280$ $21$ $21$ $16$
280.336.21.kb.1 $280$ $28$ $28$ $21$