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Properties

Label 24.96.1.bk.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
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: $0$
$\Q$-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: 24.96.1.960

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\16&19\end{bmatrix}$, $\begin{bmatrix}3&8\\20&23\end{bmatrix}$, $\begin{bmatrix}3&20\\16&21\end{bmatrix}$, $\begin{bmatrix}23&20\\20&21\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2\times D_4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.bk.2.1, 24.192.1-24.bk.2.2, 24.192.1-24.bk.2.3, 24.192.1-24.bk.2.4, 24.192.1-24.bk.2.5, 24.192.1-24.bk.2.6, 24.192.1-24.bk.2.7, 24.192.1-24.bk.2.8, 120.192.1-24.bk.2.1, 120.192.1-24.bk.2.2, 120.192.1-24.bk.2.3, 120.192.1-24.bk.2.4, 120.192.1-24.bk.2.5, 120.192.1-24.bk.2.6, 120.192.1-24.bk.2.7, 120.192.1-24.bk.2.8, 168.192.1-24.bk.2.1, 168.192.1-24.bk.2.2, 168.192.1-24.bk.2.3, 168.192.1-24.bk.2.4, 168.192.1-24.bk.2.5, 168.192.1-24.bk.2.6, 168.192.1-24.bk.2.7, 168.192.1-24.bk.2.8, 264.192.1-24.bk.2.1, 264.192.1-24.bk.2.2, 264.192.1-24.bk.2.3, 264.192.1-24.bk.2.4, 264.192.1-24.bk.2.5, 264.192.1-24.bk.2.6, 264.192.1-24.bk.2.7, 264.192.1-24.bk.2.8, 312.192.1-24.bk.2.1, 312.192.1-24.bk.2.2, 312.192.1-24.bk.2.3, 312.192.1-24.bk.2.4, 312.192.1-24.bk.2.5, 312.192.1-24.bk.2.6, 312.192.1-24.bk.2.7, 312.192.1-24.bk.2.8
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 3 x^{2} - 3 y^{2} + w^{2} $
$=$ $3 x^{2} + 3 y^{2} - z^{2} - 3 w^{2}$
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{(z^{8}+8z^{6}w^{2}+20z^{4}w^{4}+16z^{2}w^{6}+16w^{8})^{3}}{w^{8}z^{4}(z^{2}+2w^{2})^{4}(z^{2}+4w^{2})^{2}}$

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
8.48.1.m.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.k.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.m.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.t.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.v.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.w.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ba.1 $24$ $2$ $2$ $1$ $0$ 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
24.288.17.ss.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.hn.1 $24$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$