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Properties

Label 40.48.0.b.2
Level $40$
Index $48$
Genus $0$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $8$
Index: $48$ $\PSL_2$-index:$48$
Genus: $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $2$ are rational) Cusp widths $4^{8}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2^{2}\cdot4$
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: 8N0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.0.44

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&28\\0&21\end{bmatrix}$, $\begin{bmatrix}19&8\\0&27\end{bmatrix}$, $\begin{bmatrix}29&28\\2&11\end{bmatrix}$, $\begin{bmatrix}31&0\\38&1\end{bmatrix}$, $\begin{bmatrix}35&8\\14&33\end{bmatrix}$, $\begin{bmatrix}37&12\\6&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.0-40.b.2.1, 40.96.0-40.b.2.2, 40.96.0-40.b.2.3, 40.96.0-40.b.2.4, 40.96.0-40.b.2.5, 40.96.0-40.b.2.6, 40.96.0-40.b.2.7, 40.96.0-40.b.2.8, 40.96.0-40.b.2.9, 40.96.0-40.b.2.10, 40.96.0-40.b.2.11, 40.96.0-40.b.2.12, 40.96.0-40.b.2.13, 40.96.0-40.b.2.14, 40.96.0-40.b.2.15, 40.96.0-40.b.2.16, 40.96.0-40.b.2.17, 40.96.0-40.b.2.18, 40.96.0-40.b.2.19, 40.96.0-40.b.2.20, 40.96.0-40.b.2.21, 40.96.0-40.b.2.22, 40.96.0-40.b.2.23, 40.96.0-40.b.2.24, 120.96.0-40.b.2.1, 120.96.0-40.b.2.2, 120.96.0-40.b.2.3, 120.96.0-40.b.2.4, 120.96.0-40.b.2.5, 120.96.0-40.b.2.6, 120.96.0-40.b.2.7, 120.96.0-40.b.2.8, 120.96.0-40.b.2.9, 120.96.0-40.b.2.10, 120.96.0-40.b.2.11, 120.96.0-40.b.2.12, 120.96.0-40.b.2.13, 120.96.0-40.b.2.14, 120.96.0-40.b.2.15, 120.96.0-40.b.2.16, 120.96.0-40.b.2.17, 120.96.0-40.b.2.18, 120.96.0-40.b.2.19, 120.96.0-40.b.2.20, 120.96.0-40.b.2.21, 120.96.0-40.b.2.22, 120.96.0-40.b.2.23, 120.96.0-40.b.2.24, 280.96.0-40.b.2.1, 280.96.0-40.b.2.2, 280.96.0-40.b.2.3, 280.96.0-40.b.2.4, 280.96.0-40.b.2.5, 280.96.0-40.b.2.6, 280.96.0-40.b.2.7, 280.96.0-40.b.2.8, 280.96.0-40.b.2.9, 280.96.0-40.b.2.10, 280.96.0-40.b.2.11, 280.96.0-40.b.2.12, 280.96.0-40.b.2.13, 280.96.0-40.b.2.14, 280.96.0-40.b.2.15, 280.96.0-40.b.2.16, 280.96.0-40.b.2.17, 280.96.0-40.b.2.18, 280.96.0-40.b.2.19, 280.96.0-40.b.2.20, 280.96.0-40.b.2.21, 280.96.0-40.b.2.22, 280.96.0-40.b.2.23, 280.96.0-40.b.2.24
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Models

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

Rational points

This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 48 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^4}\cdot\frac{(x+y)^{48}(64x^{8}+1280x^{7}y+11040x^{6}y^{2}+53600x^{5}y^{3}+160200x^{4}y^{4}+302000x^{3}y^{5}+351500x^{2}y^{6}+232500xy^{7}+68125y^{8})^{3}(64x^{8}+1280x^{7}y+11360x^{6}y^{2}+58400x^{5}y^{3}+190200x^{4}y^{4}+402000x^{3}y^{5}+538500x^{2}y^{6}+417500xy^{7}+143125y^{8})^{3}}{y^{8}(x+y)^{48}(2x+5y)^{8}(x^{2}+5xy+5y^{2})^{4}(2x^{2}+10xy+15y^{2})^{4}(8x^{4}+80x^{3}y+300x^{2}y^{2}+500xy^{3}+325y^{4})^{4}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$
40.24.0.h.1 $40$ $2$ $2$ $0$ $0$
40.24.0.i.2 $40$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
40.96.1.a.1 $40$ $2$ $2$ $1$
40.96.1.b.2 $40$ $2$ $2$ $1$
40.96.1.e.2 $40$ $2$ $2$ $1$
40.96.1.f.1 $40$ $2$ $2$ $1$
40.96.1.l.1 $40$ $2$ $2$ $1$
40.96.1.m.2 $40$ $2$ $2$ $1$
40.96.1.n.1 $40$ $2$ $2$ $1$
40.96.1.o.2 $40$ $2$ $2$ $1$
40.96.1.p.1 $40$ $2$ $2$ $1$
40.96.1.q.2 $40$ $2$ $2$ $1$
40.96.1.w.2 $40$ $2$ $2$ $1$
40.96.1.x.1 $40$ $2$ $2$ $1$
40.96.3.r.2 $40$ $2$ $2$ $3$
40.96.3.s.2 $40$ $2$ $2$ $3$
40.96.3.u.2 $40$ $2$ $2$ $3$
40.96.3.x.1 $40$ $2$ $2$ $3$
40.240.16.c.2 $40$ $5$ $5$ $16$
40.288.15.e.1 $40$ $6$ $6$ $15$
40.480.31.g.2 $40$ $10$ $10$ $31$
120.96.1.g.2 $120$ $2$ $2$ $1$
120.96.1.h.1 $120$ $2$ $2$ $1$
120.96.1.w.1 $120$ $2$ $2$ $1$
120.96.1.x.2 $120$ $2$ $2$ $1$
120.96.1.bl.1 $120$ $2$ $2$ $1$
120.96.1.bm.2 $120$ $2$ $2$ $1$
120.96.1.bp.2 $120$ $2$ $2$ $1$
120.96.1.bq.2 $120$ $2$ $2$ $1$
120.96.1.bv.2 $120$ $2$ $2$ $1$
120.96.1.bw.1 $120$ $2$ $2$ $1$
120.96.1.cs.1 $120$ $2$ $2$ $1$
120.96.1.ct.2 $120$ $2$ $2$ $1$
120.96.3.bl.2 $120$ $2$ $2$ $3$
120.96.3.bm.2 $120$ $2$ $2$ $3$
120.96.3.by.2 $120$ $2$ $2$ $3$
120.96.3.bz.2 $120$ $2$ $2$ $3$
120.144.8.h.2 $120$ $3$ $3$ $8$
120.192.7.g.1 $120$ $4$ $4$ $7$
280.96.1.o.1 $280$ $2$ $2$ $1$
280.96.1.p.2 $280$ $2$ $2$ $1$
280.96.1.y.1 $280$ $2$ $2$ $1$
280.96.1.z.2 $280$ $2$ $2$ $1$
280.96.1.bn.2 $280$ $2$ $2$ $1$
280.96.1.bo.2 $280$ $2$ $2$ $1$
280.96.1.bp.1 $280$ $2$ $2$ $1$
280.96.1.bq.2 $280$ $2$ $2$ $1$
280.96.1.bx.2 $280$ $2$ $2$ $1$
280.96.1.by.1 $280$ $2$ $2$ $1$
280.96.1.ck.2 $280$ $2$ $2$ $1$
280.96.1.cl.1 $280$ $2$ $2$ $1$
280.96.3.ca.2 $280$ $2$ $2$ $3$
280.96.3.cb.2 $280$ $2$ $2$ $3$
280.96.3.cc.2 $280$ $2$ $2$ $3$
280.96.3.cd.2 $280$ $2$ $2$ $3$
280.384.23.g.2 $280$ $8$ $8$ $23$