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Properties

Label 24.96.1.e.1
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: 24.96.1.861

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&16\\4&19\end{bmatrix}$, $\begin{bmatrix}5&16\\16&21\end{bmatrix}$, $\begin{bmatrix}11&12\\6&13\end{bmatrix}$, $\begin{bmatrix}11&16\\14&9\end{bmatrix}$, $\begin{bmatrix}17&8\\20&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.e.1.1, 24.192.1-24.e.1.2, 24.192.1-24.e.1.3, 24.192.1-24.e.1.4, 24.192.1-24.e.1.5, 24.192.1-24.e.1.6, 24.192.1-24.e.1.7, 24.192.1-24.e.1.8, 24.192.1-24.e.1.9, 24.192.1-24.e.1.10, 24.192.1-24.e.1.11, 24.192.1-24.e.1.12, 120.192.1-24.e.1.1, 120.192.1-24.e.1.2, 120.192.1-24.e.1.3, 120.192.1-24.e.1.4, 120.192.1-24.e.1.5, 120.192.1-24.e.1.6, 120.192.1-24.e.1.7, 120.192.1-24.e.1.8, 120.192.1-24.e.1.9, 120.192.1-24.e.1.10, 120.192.1-24.e.1.11, 120.192.1-24.e.1.12, 168.192.1-24.e.1.1, 168.192.1-24.e.1.2, 168.192.1-24.e.1.3, 168.192.1-24.e.1.4, 168.192.1-24.e.1.5, 168.192.1-24.e.1.6, 168.192.1-24.e.1.7, 168.192.1-24.e.1.8, 168.192.1-24.e.1.9, 168.192.1-24.e.1.10, 168.192.1-24.e.1.11, 168.192.1-24.e.1.12, 264.192.1-24.e.1.1, 264.192.1-24.e.1.2, 264.192.1-24.e.1.3, 264.192.1-24.e.1.4, 264.192.1-24.e.1.5, 264.192.1-24.e.1.6, 264.192.1-24.e.1.7, 264.192.1-24.e.1.8, 264.192.1-24.e.1.9, 264.192.1-24.e.1.10, 264.192.1-24.e.1.11, 264.192.1-24.e.1.12, 312.192.1-24.e.1.1, 312.192.1-24.e.1.2, 312.192.1-24.e.1.3, 312.192.1-24.e.1.4, 312.192.1-24.e.1.5, 312.192.1-24.e.1.6, 312.192.1-24.e.1.7, 312.192.1-24.e.1.8, 312.192.1-24.e.1.9, 312.192.1-24.e.1.10, 312.192.1-24.e.1.11, 312.192.1-24.e.1.12
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
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 $ $=$ $ 2 y z - 3 w^{2} $
$=$ $6 x^{2} - y^{2} - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 36 x^{4} - 6 x^{2} y^{2} + z^{4} $
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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 between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 6x$
$\displaystyle Z$ $=$ $\displaystyle 3w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^5}{3^4}\cdot\frac{1024y^{24}+217728y^{16}w^{8}+15510204y^{8}w^{16}+1024z^{24}+217728z^{16}w^{8}+15510204z^{8}w^{16}+375728787w^{24}}{w^{8}(128y^{16}-2592y^{8}w^{8}+128z^{16}-2592z^{8}w^{8}+19683w^{16})}$

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.g.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.a.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.b.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.x.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.y.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.w.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.x.2 $24$ $2$ $2$ $1$ $0$ 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
24.192.5.o.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.p.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.q.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.r.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.17.kt.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.dp.1 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
120.192.5.bk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bm.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bn.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bk.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bl.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bm.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bn.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bk.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bm.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bn.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bk.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bl.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bm.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bn.1 $312$ $2$ $2$ $5$ $?$ not computed