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Properties

Label 40.48.1.o.1
Level $40$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.20

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}11&20\\8&39\end{bmatrix}$, $\begin{bmatrix}25&16\\26&3\end{bmatrix}$, $\begin{bmatrix}27&32\\34&25\end{bmatrix}$, $\begin{bmatrix}35&16\\18&21\end{bmatrix}$, $\begin{bmatrix}35&36\\28&11\end{bmatrix}$, $\begin{bmatrix}37&0\\14&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.o.1.1, 40.96.1-40.o.1.2, 40.96.1-40.o.1.3, 40.96.1-40.o.1.4, 40.96.1-40.o.1.5, 40.96.1-40.o.1.6, 40.96.1-40.o.1.7, 40.96.1-40.o.1.8, 40.96.1-40.o.1.9, 40.96.1-40.o.1.10, 40.96.1-40.o.1.11, 40.96.1-40.o.1.12, 120.96.1-40.o.1.1, 120.96.1-40.o.1.2, 120.96.1-40.o.1.3, 120.96.1-40.o.1.4, 120.96.1-40.o.1.5, 120.96.1-40.o.1.6, 120.96.1-40.o.1.7, 120.96.1-40.o.1.8, 120.96.1-40.o.1.9, 120.96.1-40.o.1.10, 120.96.1-40.o.1.11, 120.96.1-40.o.1.12, 280.96.1-40.o.1.1, 280.96.1-40.o.1.2, 280.96.1-40.o.1.3, 280.96.1-40.o.1.4, 280.96.1-40.o.1.5, 280.96.1-40.o.1.6, 280.96.1-40.o.1.7, 280.96.1-40.o.1.8, 280.96.1-40.o.1.9, 280.96.1-40.o.1.10, 280.96.1-40.o.1.11, 280.96.1-40.o.1.12
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 25x $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5^4}\cdot\frac{43750x^{2}y^{12}z^{2}+44443359375x^{2}y^{8}z^{6}+4218292236328125x^{2}y^{4}z^{10}+9763240814208984375x^{2}z^{14}+200xy^{14}z+624609375xy^{10}z^{5}+156274414062500xy^{6}z^{9}+1953220367431640625xy^{2}z^{13}+y^{16}+4812500y^{12}z^{4}+2512695312500y^{8}z^{8}+62507629394531250y^{4}z^{12}+59604644775390625z^{16}}{z^{2}y^{8}(x^{2}y^{4}+234375x^{2}z^{4}+10625xy^{2}z^{3}+150y^{4}z^{2}+390625z^{6})}$

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 Kernel decomposition
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.k.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dp.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dv.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.a.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.24.1.dj.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.24.1.dp.1 $40$ $2$ $2$ $1$ $1$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.96.1.o.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.o.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.3.m.1 $40$ $2$ $2$ $3$ $1$ $2$
40.96.3.o.1 $40$ $2$ $2$ $3$ $1$ $2$
40.96.3.v.1 $40$ $2$ $2$ $3$ $2$ $1^{2}$
40.96.3.w.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.96.3.bb.1 $40$ $2$ $2$ $3$ $1$ $2$
40.96.3.bf.1 $40$ $2$ $2$ $3$ $1$ $2$
40.240.17.y.2 $40$ $5$ $5$ $17$ $4$ $1^{14}\cdot2$
40.288.17.bv.2 $40$ $6$ $6$ $17$ $4$ $1^{14}\cdot2$
40.480.33.ew.2 $40$ $10$ $10$ $33$ $6$ $1^{28}\cdot2^{2}$
120.96.1.bo.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bo.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.u.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bx.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.cj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.cv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.144.9.cz.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bp.2 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.bm.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.bm.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.3.y.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.be.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.bw.2 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.bx.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.cl.1 $280$ $2$ $2$ $3$ $?$ not computed
280.96.3.cr.1 $280$ $2$ $2$ $3$ $?$ not computed