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Properties

Label 24.96.1.a.2
Level $24$
Index $96$
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: $96$ $\PSL_2$-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 $2^{2}\cdot4^{3}$
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: 24.96.1.303

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&2\\16&15\end{bmatrix}$, $\begin{bmatrix}7&6\\4&5\end{bmatrix}$, $\begin{bmatrix}9&4\\16&1\end{bmatrix}$, $\begin{bmatrix}13&4\\4&5\end{bmatrix}$, $\begin{bmatrix}17&2\\20&15\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.a.2.1, 24.192.1-24.a.2.2, 24.192.1-24.a.2.3, 24.192.1-24.a.2.4, 24.192.1-24.a.2.5, 24.192.1-24.a.2.6, 24.192.1-24.a.2.7, 24.192.1-24.a.2.8, 24.192.1-24.a.2.9, 24.192.1-24.a.2.10, 24.192.1-24.a.2.11, 24.192.1-24.a.2.12, 120.192.1-24.a.2.1, 120.192.1-24.a.2.2, 120.192.1-24.a.2.3, 120.192.1-24.a.2.4, 120.192.1-24.a.2.5, 120.192.1-24.a.2.6, 120.192.1-24.a.2.7, 120.192.1-24.a.2.8, 120.192.1-24.a.2.9, 120.192.1-24.a.2.10, 120.192.1-24.a.2.11, 120.192.1-24.a.2.12, 168.192.1-24.a.2.1, 168.192.1-24.a.2.2, 168.192.1-24.a.2.3, 168.192.1-24.a.2.4, 168.192.1-24.a.2.5, 168.192.1-24.a.2.6, 168.192.1-24.a.2.7, 168.192.1-24.a.2.8, 168.192.1-24.a.2.9, 168.192.1-24.a.2.10, 168.192.1-24.a.2.11, 168.192.1-24.a.2.12, 264.192.1-24.a.2.1, 264.192.1-24.a.2.2, 264.192.1-24.a.2.3, 264.192.1-24.a.2.4, 264.192.1-24.a.2.5, 264.192.1-24.a.2.6, 264.192.1-24.a.2.7, 264.192.1-24.a.2.8, 264.192.1-24.a.2.9, 264.192.1-24.a.2.10, 264.192.1-24.a.2.11, 264.192.1-24.a.2.12, 312.192.1-24.a.2.1, 312.192.1-24.a.2.2, 312.192.1-24.a.2.3, 312.192.1-24.a.2.4, 312.192.1-24.a.2.5, 312.192.1-24.a.2.6, 312.192.1-24.a.2.7, 312.192.1-24.a.2.8, 312.192.1-24.a.2.9, 312.192.1-24.a.2.10, 312.192.1-24.a.2.11, 312.192.1-24.a.2.12
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $768$

Jacobian

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

Models

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

$ 0 $ $=$ $ x^{2} + y^{2} + z^{2} $
$=$ $3 x^{2} - 3 y^{2} - 2 w^{2}$
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Rational points

This modular curve has no real points, and therefore no 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^4}\cdot\frac{(9z^{4}-18z^{3}w+18z^{2}w^{2}-12zw^{3}+4w^{4})^{3}(9z^{4}+18z^{3}w+18z^{2}w^{2}+12zw^{3}+4w^{4})^{3}}{w^{8}z^{8}(3z^{2}-2w^{2})^{2}(3z^{2}+2w^{2})^{2}}$

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
8.48.0.a.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.b.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.p.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.q.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.n.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bg.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.48.1.bh.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
24.192.5.i.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.k.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.l.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.n.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.17.kf.1 $24$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
24.384.17.dd.1 $24$ $4$ $4$ $17$ $3$ $1^{8}\cdot2^{4}$
120.192.5.p.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.q.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.u.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.w.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.p.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.q.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.u.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.w.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.p.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.q.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.u.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.w.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.p.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.q.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.u.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.w.1 $312$ $2$ $2$ $5$ $?$ not computed