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Properties

Label 10.24.1.a.2
Level $10$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $10$ $\SL_2$-level: $10$ Newform level: $20$
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: $0$
$\Q$-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: 10.24.1.3

Level structure

$\GL_2(\Z/10\Z)$-generators: $\begin{bmatrix}1&5\\9&4\end{bmatrix}$, $\begin{bmatrix}6&1\\5&3\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup: $C_6\times F_5$
Contains $-I$: yes
Quadratic refinements: 10.48.1-10.a.2.1, 10.48.1-10.a.2.2, 20.48.1-10.a.2.1, 20.48.1-10.a.2.2, 20.48.1-10.a.2.3, 20.48.1-10.a.2.4, 30.48.1-10.a.2.1, 30.48.1-10.a.2.2, 40.48.1-10.a.2.1, 40.48.1-10.a.2.2, 40.48.1-10.a.2.3, 40.48.1-10.a.2.4, 40.48.1-10.a.2.5, 40.48.1-10.a.2.6, 60.48.1-10.a.2.1, 60.48.1-10.a.2.2, 60.48.1-10.a.2.3, 60.48.1-10.a.2.4, 70.48.1-10.a.2.1, 70.48.1-10.a.2.2, 110.48.1-10.a.2.1, 110.48.1-10.a.2.2, 120.48.1-10.a.2.1, 120.48.1-10.a.2.2, 120.48.1-10.a.2.3, 120.48.1-10.a.2.4, 120.48.1-10.a.2.5, 120.48.1-10.a.2.6, 130.48.1-10.a.2.1, 130.48.1-10.a.2.2, 140.48.1-10.a.2.1, 140.48.1-10.a.2.2, 140.48.1-10.a.2.3, 140.48.1-10.a.2.4, 170.48.1-10.a.2.1, 170.48.1-10.a.2.2, 190.48.1-10.a.2.1, 190.48.1-10.a.2.2, 210.48.1-10.a.2.1, 210.48.1-10.a.2.2, 220.48.1-10.a.2.1, 220.48.1-10.a.2.2, 220.48.1-10.a.2.3, 220.48.1-10.a.2.4, 230.48.1-10.a.2.1, 230.48.1-10.a.2.2, 260.48.1-10.a.2.1, 260.48.1-10.a.2.2, 260.48.1-10.a.2.3, 260.48.1-10.a.2.4, 280.48.1-10.a.2.1, 280.48.1-10.a.2.2, 280.48.1-10.a.2.3, 280.48.1-10.a.2.4, 280.48.1-10.a.2.5, 280.48.1-10.a.2.6, 290.48.1-10.a.2.1, 290.48.1-10.a.2.2, 310.48.1-10.a.2.1, 310.48.1-10.a.2.2, 330.48.1-10.a.2.1, 330.48.1-10.a.2.2
Cyclic 10-isogeny field degree: $3$
Cyclic 10-torsion field degree: $12$
Full 10-torsion field degree: $120$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 41x - 116 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(-4:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{728x^{2}y^{6}+615053640x^{2}y^{4}z^{2}+1564941405544x^{2}y^{2}z^{4}+457916259765624x^{2}z^{6}+186560xy^{6}z+12246636672xy^{4}z^{3}+16353759767744xy^{2}z^{5}+3704620361328128xz^{7}+y^{8}+18969452y^{6}z^{2}+150229845670y^{4}z^{4}+81666259786092y^{2}z^{6}+7491821289062529z^{8}}{y^{2}(x^{2}y^{4}+22x^{2}y^{2}z^{2}-x^{2}z^{4}-14xy^{4}z-65xy^{2}z^{3}+3xz^{5}+47y^{4}z^{2}-645y^{2}z^{4}+29z^{6})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}(2)$ $2$ $12$ $12$ $0$ $0$ full Jacobian
5.12.0.a.2 $5$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
5.12.0.a.2 $5$ $2$ $2$ $0$ $0$ full Jacobian
10.12.0.a.2 $10$ $2$ $2$ $0$ $0$ full Jacobian
10.12.1.a.1 $10$ $2$ $2$ $1$ $0$ 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
10.72.1.a.2 $10$ $3$ $3$ $1$ $0$ dimension zero
10.120.5.c.1 $10$ $5$ $5$ $5$ $0$ $1^{2}\cdot2$
20.48.3.a.1 $20$ $2$ $2$ $3$ $0$ $2$
20.48.3.c.2 $20$ $2$ $2$ $3$ $0$ $2$
20.96.5.a.2 $20$ $4$ $4$ $5$ $0$ $1^{2}\cdot2$
30.72.5.a.1 $30$ $3$ $3$ $5$ $0$ $1^{2}\cdot2$
30.96.5.a.1 $30$ $4$ $4$ $5$ $0$ $1^{2}\cdot2$
40.48.3.a.1 $40$ $2$ $2$ $3$ $0$ $2$
40.48.3.c.2 $40$ $2$ $2$ $3$ $0$ $2$
50.120.5.a.2 $50$ $5$ $5$ $5$ $0$ $1^{2}\cdot2$
60.48.3.d.2 $60$ $2$ $2$ $3$ $0$ $2$
60.48.3.f.1 $60$ $2$ $2$ $3$ $0$ $2$
70.72.1.b.1 $70$ $3$ $3$ $1$ $0$ dimension zero
70.192.13.a.1 $70$ $8$ $8$ $13$ $0$ $1^{4}\cdot2^{4}$
70.504.37.a.2 $70$ $21$ $21$ $37$ $4$ $1^{4}\cdot2^{8}\cdot4^{4}$
70.672.49.a.2 $70$ $28$ $28$ $49$ $4$ $1^{8}\cdot2^{12}\cdot4^{4}$
90.72.1.a.2 $90$ $3$ $3$ $1$ $?$ dimension zero
110.288.21.a.1 $110$ $12$ $12$ $21$ $?$ not computed
120.48.3.e.2 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.g.1 $120$ $2$ $2$ $3$ $?$ not computed
130.72.1.b.2 $130$ $3$ $3$ $1$ $?$ dimension zero
140.48.3.a.1 $140$ $2$ $2$ $3$ $?$ not computed
140.48.3.c.2 $140$ $2$ $2$ $3$ $?$ not computed
190.72.1.b.2 $190$ $3$ $3$ $1$ $?$ dimension zero
220.48.3.a.2 $220$ $2$ $2$ $3$ $?$ not computed
220.48.3.c.2 $220$ $2$ $2$ $3$ $?$ not computed
260.48.3.a.2 $260$ $2$ $2$ $3$ $?$ not computed
260.48.3.c.1 $260$ $2$ $2$ $3$ $?$ not computed
280.48.3.a.1 $280$ $2$ $2$ $3$ $?$ not computed
280.48.3.c.2 $280$ $2$ $2$ $3$ $?$ not computed
310.72.1.b.2 $310$ $3$ $3$ $1$ $?$ dimension zero