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Properties

Label 40.24.1.cp.1
Level $40$
Index $24$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $1600$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot10^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.24.1.20

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&36\\11&35\end{bmatrix}$, $\begin{bmatrix}10&39\\33&36\end{bmatrix}$, $\begin{bmatrix}12&17\\39&15\end{bmatrix}$, $\begin{bmatrix}18&21\\39&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.48.1-40.cp.1.1, 40.48.1-40.cp.1.2, 40.48.1-40.cp.1.3, 40.48.1-40.cp.1.4, 40.48.1-40.cp.1.5, 40.48.1-40.cp.1.6, 40.48.1-40.cp.1.7, 40.48.1-40.cp.1.8, 120.48.1-40.cp.1.1, 120.48.1-40.cp.1.2, 120.48.1-40.cp.1.3, 120.48.1-40.cp.1.4, 120.48.1-40.cp.1.5, 120.48.1-40.cp.1.6, 120.48.1-40.cp.1.7, 120.48.1-40.cp.1.8, 280.48.1-40.cp.1.1, 280.48.1-40.cp.1.2, 280.48.1-40.cp.1.3, 280.48.1-40.cp.1.4, 280.48.1-40.cp.1.5, 280.48.1-40.cp.1.6, 280.48.1-40.cp.1.7, 280.48.1-40.cp.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $30720$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 4133x - 103637 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^5\cdot5^5}\cdot\frac{80x^{2}y^{6}+15600000x^{2}y^{4}z^{2}+942640000000x^{2}y^{2}z^{4}+18756240000000000x^{2}z^{6}+7520xy^{6}z+1245600000xy^{4}z^{3}+72102560000000xy^{2}z^{5}+1404875360000000000xz^{7}+y^{8}+308720y^{6}z^{2}+33152800000y^{4}z^{4}+1546420560000000y^{2}z^{6}+26303096760000000000z^{8}}{z^{3}y^{2}(12200x^{2}z+xy^{2}+913800xz^{2}+147y^{2}z+17108800z^{3})}$

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
$X_{\pm1}(5)$ $5$ $2$ $2$ $0$ $0$ full Jacobian
40.12.0.bw.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.12.1.d.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
40.72.1.bf.1 $40$ $3$ $3$ $1$ $1$ dimension zero
40.96.5.t.2 $40$ $4$ $4$ $5$ $1$ $1^{2}\cdot2$
40.120.5.cu.1 $40$ $5$ $5$ $5$ $2$ $1^{2}\cdot2$
120.72.5.bbf.2 $120$ $3$ $3$ $5$ $?$ not computed
120.96.5.nj.2 $120$ $4$ $4$ $5$ $?$ not computed
200.120.5.t.1 $200$ $5$ $5$ $5$ $?$ not computed
280.192.13.fv.2 $280$ $8$ $8$ $13$ $?$ not computed