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Properties

Label 24.192.9.if.1
Level $24$
Index $192$
Genus $9$
Analytic rank $3$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $576$
Index: $192$ $\PSL_2$-index:$192$
Genus: $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}\cdot12^{4}\cdot24^{4}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $3$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24AF9
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.9.347

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&10\\18&5\end{bmatrix}$, $\begin{bmatrix}11&13\\18&13\end{bmatrix}$, $\begin{bmatrix}13&17\\6&7\end{bmatrix}$, $\begin{bmatrix}17&4\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^5.D_6$
Contains $-I$: yes
Quadratic refinements: 24.384.9-24.if.1.1, 24.384.9-24.if.1.2, 24.384.9-24.if.1.3, 24.384.9-24.if.1.4, 24.384.9-24.if.1.5, 24.384.9-24.if.1.6, 24.384.9-24.if.1.7, 24.384.9-24.if.1.8, 48.384.9-24.if.1.1, 48.384.9-24.if.1.2, 48.384.9-24.if.1.3, 48.384.9-24.if.1.4, 120.384.9-24.if.1.1, 120.384.9-24.if.1.2, 120.384.9-24.if.1.3, 120.384.9-24.if.1.4, 120.384.9-24.if.1.5, 120.384.9-24.if.1.6, 120.384.9-24.if.1.7, 120.384.9-24.if.1.8, 168.384.9-24.if.1.1, 168.384.9-24.if.1.2, 168.384.9-24.if.1.3, 168.384.9-24.if.1.4, 168.384.9-24.if.1.5, 168.384.9-24.if.1.6, 168.384.9-24.if.1.7, 168.384.9-24.if.1.8, 240.384.9-24.if.1.1, 240.384.9-24.if.1.2, 240.384.9-24.if.1.3, 240.384.9-24.if.1.4, 264.384.9-24.if.1.1, 264.384.9-24.if.1.2, 264.384.9-24.if.1.3, 264.384.9-24.if.1.4, 264.384.9-24.if.1.5, 264.384.9-24.if.1.6, 264.384.9-24.if.1.7, 264.384.9-24.if.1.8, 312.384.9-24.if.1.1, 312.384.9-24.if.1.2, 312.384.9-24.if.1.3, 312.384.9-24.if.1.4, 312.384.9-24.if.1.5, 312.384.9-24.if.1.6, 312.384.9-24.if.1.7, 312.384.9-24.if.1.8
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{44}\cdot3^{15}$
Simple: no
Squarefree: no
Decomposition: $1^{9}$
Newforms: 24.2.a.a, 48.2.a.a, 72.2.a.a, 192.2.a.b, 288.2.a.b, 288.2.a.c, 576.2.a.c$^{2}$, 576.2.a.d

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $ $=$ $ x u + x v + y s + z s + v r $
$=$ $x t + y s + z s + t r + u r$
$=$ $x t + x u + x v - y s + z s + w s$
$=$ $x s - y t - z t - z u + w u + r s$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{16} + 160 x^{14} z^{2} + 34 x^{12} y^{2} z^{2} + 8704 x^{12} z^{4} - 200 x^{10} y^{2} z^{4} + \cdots + 746496 z^{16} $
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Rational points

This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 2z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}t$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.5.bj.1 :

$\displaystyle X$ $=$ $\displaystyle -x$
$\displaystyle Y$ $=$ $\displaystyle -y$
$\displaystyle Z$ $=$ $\displaystyle -t$
$\displaystyle W$ $=$ $\displaystyle -z+w$
$\displaystyle T$ $=$ $\displaystyle u+v$

Equation of the image curve:

$0$ $=$ $ 2YW-ZT $
$=$ $ 3X^{2}-YW $
$=$ $ 18Y^{2}+Z^{2}-2W^{2}-T^{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
24.48.1.dm.1 $24$ $4$ $4$ $1$ $1$ $1^{8}$
24.96.3.dh.1 $24$ $2$ $2$ $3$ $0$ $1^{6}$
24.96.3.el.1 $24$ $2$ $2$ $3$ $0$ $1^{6}$
24.96.5.bj.1 $24$ $2$ $2$ $5$ $3$ $1^{4}$
24.96.5.dx.1 $24$ $2$ $2$ $5$ $2$ $1^{4}$
24.96.5.dy.1 $24$ $2$ $2$ $5$ $2$ $1^{4}$
24.96.5.ge.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.96.5.gh.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.576.33.byu.1 $24$ $3$ $3$ $33$ $8$ $1^{24}$
48.384.21.qy.1 $48$ $2$ $2$ $21$ $3$ $2^{2}\cdot4^{2}$
48.384.21.qy.2 $48$ $2$ $2$ $21$ $3$ $2^{2}\cdot4^{2}$
48.384.21.vi.1 $48$ $2$ $2$ $21$ $3$ $2^{2}\cdot4^{2}$
48.384.21.vi.2 $48$ $2$ $2$ $21$ $3$ $2^{2}\cdot4^{2}$
240.384.21.cnn.1 $240$ $2$ $2$ $21$ $?$ not computed
240.384.21.cnn.2 $240$ $2$ $2$ $21$ $?$ not computed
240.384.21.cxt.1 $240$ $2$ $2$ $21$ $?$ not computed
240.384.21.cxt.2 $240$ $2$ $2$ $21$ $?$ not computed