Properties

Label 24.192.1-24.f.1.1
Level $24$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $32$
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: $0$
$\Q$-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: 24.192.1.682

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&16\\8&9\end{bmatrix}$, $\begin{bmatrix}13&20\\8&21\end{bmatrix}$, $\begin{bmatrix}15&16\\8&13\end{bmatrix}$, $\begin{bmatrix}19&8\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^3\times \GL(2,3)$
Contains $-I$: no $\quad$ (see 24.96.1.f.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 3 x^{2} + 3 y^{2} - w^{2} $
$=$ $3 x^{2} - 3 y^{2} + z^{2}$
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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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(z^{8}-z^{4}w^{4}+w^{8})^{3}}{w^{8}z^{8}(z-w)^{2}(z+w)^{2}(z^{2}+w^{2})^{2}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1-8.h.1.2 $8$ $2$ $2$ $1$ $0$ dimension zero
24.96.0-24.a.1.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.a.1.4 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.b.2.9 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.b.2.13 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.ba.1.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.ba.1.12 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.bb.1.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.bb.1.14 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.96.1-8.h.1.12 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1-24.m.2.6 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1-24.m.2.9 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1-24.q.1.5 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1-24.q.1.12 $24$ $2$ $2$ $1$ $0$ 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
24.384.5-24.r.2.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.r.3.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.384.5-24.s.2.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.384.5-24.s.4.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.576.17-24.ku.2.5 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.768.17-24.dq.1.17 $24$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
48.384.5-48.y.3.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.384.5-48.y.4.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.384.5-48.z.3.4 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.384.5-48.z.4.4 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.384.9-48.dn.1.9 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
48.384.9-48.dn.2.9 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
48.384.9-48.do.1.9 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
48.384.9-48.do.2.9 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
120.384.5-120.bo.2.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.bo.4.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.bp.2.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.bp.3.9 $120$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.bo.3.3 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.bo.4.1 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.bp.3.1 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.bp.4.5 $168$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.gg.1.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.gg.2.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.gh.1.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.gh.2.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.9-240.sj.3.17 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.sj.4.17 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.sk.3.17 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.sk.4.17 $240$ $2$ $2$ $9$ $?$ not computed
264.384.5-264.bo.3.3 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.bo.4.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.bp.3.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.bp.4.5 $264$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.bo.3.3 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.bo.4.1 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.bp.3.1 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.bp.4.5 $312$ $2$ $2$ $5$ $?$ not computed