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Properties

Label 40.96.0-8.c.1.4
Level $40$
Index $96$
Genus $0$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $4$

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Invariants

Level: $40$ $\SL_2$-level: $8$
Index: $96$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{2}$ Cusp orbits $1^{4}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8N0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.96.0.229

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&4\\4&23\end{bmatrix}$, $\begin{bmatrix}9&4\\36&25\end{bmatrix}$, $\begin{bmatrix}15&4\\32&9\end{bmatrix}$, $\begin{bmatrix}31&20\\28&3\end{bmatrix}$, $\begin{bmatrix}39&36\\24&27\end{bmatrix}$
Contains $-I$: no $\quad$ (see 8.48.0.c.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

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 48 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{48}(x^{8}-4x^{7}y+4x^{6}y^{2}+28x^{5}y^{3}+6x^{4}y^{4}-28x^{3}y^{5}+4x^{2}y^{6}+4xy^{7}+y^{8})^{3}(x^{8}+4x^{7}y+4x^{6}y^{2}-28x^{5}y^{3}+6x^{4}y^{4}+28x^{3}y^{5}+4x^{2}y^{6}-4xy^{7}+y^{8})^{3}}{y^{4}x^{52}(x-y)^{4}(x+y)^{4}(x^{2}+y^{2})^{8}(x^{2}-2xy-y^{2})^{4}(x^{2}+2xy-y^{2})^{4}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
40.48.0-4.b.1.8 $40$ $2$ $2$ $0$ $0$
40.48.0-4.b.1.9 $40$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
40.192.1-8.f.1.4 $40$ $2$ $2$ $1$
40.192.1-8.f.2.5 $40$ $2$ $2$ $1$
40.192.1-8.g.1.2 $40$ $2$ $2$ $1$
40.192.1-8.g.2.4 $40$ $2$ $2$ $1$
40.192.3-8.i.1.1 $40$ $2$ $2$ $3$
40.192.3-8.i.1.6 $40$ $2$ $2$ $3$
40.192.3-8.j.1.3 $40$ $2$ $2$ $3$
40.192.3-8.j.1.5 $40$ $2$ $2$ $3$
80.192.2-16.a.1.7 $80$ $2$ $2$ $2$
80.192.2-16.a.1.8 $80$ $2$ $2$ $2$
80.192.2-16.b.1.13 $80$ $2$ $2$ $2$
80.192.2-16.b.1.15 $80$ $2$ $2$ $2$
80.192.2-16.c.1.7 $80$ $2$ $2$ $2$
80.192.2-16.c.1.15 $80$ $2$ $2$ $2$
80.192.2-16.d.1.4 $80$ $2$ $2$ $2$
80.192.2-16.d.1.8 $80$ $2$ $2$ $2$
120.192.1-24.w.1.9 $120$ $2$ $2$ $1$
120.192.1-24.w.2.9 $120$ $2$ $2$ $1$
120.192.1-24.x.1.11 $120$ $2$ $2$ $1$
120.192.1-24.x.2.9 $120$ $2$ $2$ $1$
120.192.3-24.z.1.5 $120$ $2$ $2$ $3$
120.192.3-24.z.1.8 $120$ $2$ $2$ $3$
120.192.3-24.ba.1.2 $120$ $2$ $2$ $3$
120.192.3-24.ba.1.3 $120$ $2$ $2$ $3$
120.288.8-24.l.1.39 $120$ $3$ $3$ $8$
120.384.7-24.i.1.31 $120$ $4$ $4$ $7$
40.192.1-40.w.1.11 $40$ $2$ $2$ $1$
40.192.1-40.w.2.10 $40$ $2$ $2$ $1$
40.192.1-40.x.1.9 $40$ $2$ $2$ $1$
40.192.1-40.x.2.9 $40$ $2$ $2$ $1$
40.192.3-40.be.1.1 $40$ $2$ $2$ $3$
40.192.3-40.be.1.11 $40$ $2$ $2$ $3$
40.192.3-40.bf.1.7 $40$ $2$ $2$ $3$
40.192.3-40.bf.1.10 $40$ $2$ $2$ $3$
40.480.16-40.f.1.12 $40$ $5$ $5$ $16$
40.576.15-40.i.1.36 $40$ $6$ $6$ $15$
40.960.31-40.l.1.34 $40$ $10$ $10$ $31$
240.192.2-48.a.1.27 $240$ $2$ $2$ $2$
240.192.2-48.a.1.32 $240$ $2$ $2$ $2$
240.192.2-48.b.1.21 $240$ $2$ $2$ $2$
240.192.2-48.b.1.30 $240$ $2$ $2$ $2$
240.192.2-48.c.1.9 $240$ $2$ $2$ $2$
240.192.2-48.c.1.30 $240$ $2$ $2$ $2$
240.192.2-48.d.1.13 $240$ $2$ $2$ $2$
240.192.2-48.d.1.32 $240$ $2$ $2$ $2$
280.192.1-56.w.1.11 $280$ $2$ $2$ $1$
280.192.1-56.w.2.10 $280$ $2$ $2$ $1$
280.192.1-56.x.1.11 $280$ $2$ $2$ $1$
280.192.1-56.x.2.10 $280$ $2$ $2$ $1$
280.192.3-56.w.1.7 $280$ $2$ $2$ $3$
280.192.3-56.w.1.12 $280$ $2$ $2$ $3$
280.192.3-56.x.1.5 $280$ $2$ $2$ $3$
280.192.3-56.x.1.9 $280$ $2$ $2$ $3$
80.192.2-80.e.1.19 $80$ $2$ $2$ $2$
80.192.2-80.e.1.31 $80$ $2$ $2$ $2$
80.192.2-80.f.1.20 $80$ $2$ $2$ $2$
80.192.2-80.f.1.23 $80$ $2$ $2$ $2$
80.192.2-80.g.1.20 $80$ $2$ $2$ $2$
80.192.2-80.g.1.23 $80$ $2$ $2$ $2$
80.192.2-80.h.1.25 $80$ $2$ $2$ $2$
80.192.2-80.h.1.31 $80$ $2$ $2$ $2$
120.192.1-120.cy.1.19 $120$ $2$ $2$ $1$
120.192.1-120.cy.2.19 $120$ $2$ $2$ $1$
120.192.1-120.cz.1.19 $120$ $2$ $2$ $1$
120.192.1-120.cz.2.22 $120$ $2$ $2$ $1$
120.192.3-120.cw.1.7 $120$ $2$ $2$ $3$
120.192.3-120.cw.1.21 $120$ $2$ $2$ $3$
120.192.3-120.cx.1.6 $120$ $2$ $2$ $3$
120.192.3-120.cx.1.22 $120$ $2$ $2$ $3$
240.192.2-240.e.1.46 $240$ $2$ $2$ $2$
240.192.2-240.e.1.59 $240$ $2$ $2$ $2$
240.192.2-240.f.1.38 $240$ $2$ $2$ $2$
240.192.2-240.f.1.63 $240$ $2$ $2$ $2$
240.192.2-240.g.1.45 $240$ $2$ $2$ $2$
240.192.2-240.g.1.56 $240$ $2$ $2$ $2$
240.192.2-240.h.1.47 $240$ $2$ $2$ $2$
240.192.2-240.h.1.52 $240$ $2$ $2$ $2$
280.192.1-280.cy.1.22 $280$ $2$ $2$ $1$
280.192.1-280.cy.2.24 $280$ $2$ $2$ $1$
280.192.1-280.cz.1.18 $280$ $2$ $2$ $1$
280.192.1-280.cz.2.24 $280$ $2$ $2$ $1$
280.192.3-280.cw.1.3 $280$ $2$ $2$ $3$
280.192.3-280.cw.1.23 $280$ $2$ $2$ $3$
280.192.3-280.cx.1.12 $280$ $2$ $2$ $3$
280.192.3-280.cx.1.23 $280$ $2$ $2$ $3$