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Properties

Label 24.96.3.do.2
Level $24$
Index $96$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24U3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.3.630

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&9\\14&5\end{bmatrix}$, $\begin{bmatrix}13&0\\18&11\end{bmatrix}$, $\begin{bmatrix}17&12\\22&23\end{bmatrix}$, $\begin{bmatrix}23&15\\0&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.135352
Contains $-I$: yes
Quadratic refinements: 24.192.3-24.do.2.1, 24.192.3-24.do.2.2, 24.192.3-24.do.2.3, 24.192.3-24.do.2.4, 24.192.3-24.do.2.5, 24.192.3-24.do.2.6, 24.192.3-24.do.2.7, 24.192.3-24.do.2.8, 48.192.3-24.do.2.1, 48.192.3-24.do.2.2, 48.192.3-24.do.2.3, 48.192.3-24.do.2.4, 48.192.3-24.do.2.5, 48.192.3-24.do.2.6, 48.192.3-24.do.2.7, 48.192.3-24.do.2.8, 120.192.3-24.do.2.1, 120.192.3-24.do.2.2, 120.192.3-24.do.2.3, 120.192.3-24.do.2.4, 120.192.3-24.do.2.5, 120.192.3-24.do.2.6, 120.192.3-24.do.2.7, 120.192.3-24.do.2.8, 168.192.3-24.do.2.1, 168.192.3-24.do.2.2, 168.192.3-24.do.2.3, 168.192.3-24.do.2.4, 168.192.3-24.do.2.5, 168.192.3-24.do.2.6, 168.192.3-24.do.2.7, 168.192.3-24.do.2.8, 240.192.3-24.do.2.1, 240.192.3-24.do.2.2, 240.192.3-24.do.2.3, 240.192.3-24.do.2.4, 240.192.3-24.do.2.5, 240.192.3-24.do.2.6, 240.192.3-24.do.2.7, 240.192.3-24.do.2.8, 264.192.3-24.do.2.1, 264.192.3-24.do.2.2, 264.192.3-24.do.2.3, 264.192.3-24.do.2.4, 264.192.3-24.do.2.5, 264.192.3-24.do.2.6, 264.192.3-24.do.2.7, 264.192.3-24.do.2.8, 312.192.3-24.do.2.1, 312.192.3-24.do.2.2, 312.192.3-24.do.2.3, 312.192.3-24.do.2.4, 312.192.3-24.do.2.5, 312.192.3-24.do.2.6, 312.192.3-24.do.2.7, 312.192.3-24.do.2.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{18}\cdot3^{3}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 192.2.a.b, 192.2.c.a

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

$ 0 $ $=$ $ 2 x^{2} + t^{2} - u^{2} $
$=$ $2 y w - 2 z w + z u - w t - t^{2} + t u$
$=$ $2 y z - 2 y t - 2 z^{2} - z t + w t$
$=$ $2 y z + 2 y w - y t + z^{2} + z w - 2 z t + z u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 2 x^{2} z^{6} + 144 y^{8} + 40 y^{4} z^{4} + z^{8} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ 2x^{8} + 80x^{4} + 288 $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{4}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{3}{4}t$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle t$
$\displaystyle Y$ $=$ $\displaystyle -\frac{8}{3}xt^{3}$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{(t-2u)^{3}(11t^{3}-6t^{2}u-12tu^{2}+8u^{3})^{3}}{t^{6}(t-u)^{3}(t+u)(5t-4u)^{2}}$

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
24.48.0.br.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.dq.1 $24$ $2$ $2$ $1$ $0$ $2$
24.48.2.e.2 $24$ $2$ $2$ $2$ $0$ $1$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.13.hh.1 $24$ $3$ $3$ $13$ $2$ $1^{4}\cdot2^{3}$
48.192.9.os.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.ot.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.ou.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.ov.2 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.pr.1 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.ps.1 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.pt.1 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
48.192.9.pu.1 $48$ $2$ $2$ $9$ $0$ $2\cdot4$
72.288.13.cn.2 $72$ $3$ $3$ $13$ $?$ not computed
72.288.19.ew.1 $72$ $3$ $3$ $19$ $?$ not computed
72.288.19.fd.1 $72$ $3$ $3$ $19$ $?$ not computed
240.192.9.cmi.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cmj.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cmk.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cml.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cob.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.coc.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cod.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.coe.1 $240$ $2$ $2$ $9$ $?$ not computed