Properties

Label 24.192.1-24.n.1.6
Level $24$
Index $192$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $576$
Index: $192$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.1.6

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&16\\4&9\end{bmatrix}$, $\begin{bmatrix}5&12\\12&5\end{bmatrix}$, $\begin{bmatrix}21&16\\4&13\end{bmatrix}$, $\begin{bmatrix}23&20\\0&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^3\times \GL(2,3)$
Contains $-I$: no $\quad$ (see 24.96.1.n.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 9x $
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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 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 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Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.0-8.b.2.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-8.b.2.9 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.b.2.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.b.2.8 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.1-24.n.1.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1-24.n.1.2 $24$ $2$ $2$ $1$ $1$ 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
24.384.5-24.l.1.6 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.384.5-24.n.2.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.384.5-24.p.1.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.384.5-24.s.2.6 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.384.5-24.x.1.6 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.384.5-24.ba.3.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.384.5-24.bh.1.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.384.5-24.bj.3.6 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.576.17-24.mz.2.14 $24$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
24.768.17-24.er.2.14 $24$ $4$ $4$ $17$ $3$ $1^{8}\cdot2^{4}$
120.384.5-120.cs.1.11 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ct.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.dv.1.11 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.dw.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.fb.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.fc.1.11 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.gc.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.gd.1.11 $120$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.cs.1.11 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.ct.1.4 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.dv.1.4 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.dw.1.11 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.fb.1.11 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.fc.1.4 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.gc.1.4 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.gd.1.11 $168$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.cs.1.11 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.ct.1.4 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.dv.1.4 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.dw.1.11 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.fb.1.11 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.fc.1.4 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.gc.1.4 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.gd.1.11 $264$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.cs.1.11 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.ct.1.4 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.dv.1.4 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.dw.1.11 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.fb.1.11 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.fc.1.4 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.gc.1.4 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.gd.1.11 $312$ $2$ $2$ $5$ $?$ not computed