↑

Properties

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

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
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 $4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.580

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&20\\4&17\end{bmatrix}$, $\begin{bmatrix}13&16\\4&11\end{bmatrix}$, $\begin{bmatrix}17&22\\4&13\end{bmatrix}$, $\begin{bmatrix}23&12\\16&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2\times D_4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.bt.2.1, 24.192.1-24.bt.2.2, 24.192.1-24.bt.2.3, 24.192.1-24.bt.2.4, 24.192.1-24.bt.2.5, 24.192.1-24.bt.2.6, 24.192.1-24.bt.2.7, 24.192.1-24.bt.2.8, 120.192.1-24.bt.2.1, 120.192.1-24.bt.2.2, 120.192.1-24.bt.2.3, 120.192.1-24.bt.2.4, 120.192.1-24.bt.2.5, 120.192.1-24.bt.2.6, 120.192.1-24.bt.2.7, 120.192.1-24.bt.2.8, 168.192.1-24.bt.2.1, 168.192.1-24.bt.2.2, 168.192.1-24.bt.2.3, 168.192.1-24.bt.2.4, 168.192.1-24.bt.2.5, 168.192.1-24.bt.2.6, 168.192.1-24.bt.2.7, 168.192.1-24.bt.2.8, 264.192.1-24.bt.2.1, 264.192.1-24.bt.2.2, 264.192.1-24.bt.2.3, 264.192.1-24.bt.2.4, 264.192.1-24.bt.2.5, 264.192.1-24.bt.2.6, 264.192.1-24.bt.2.7, 264.192.1-24.bt.2.8, 312.192.1-24.bt.2.1, 312.192.1-24.bt.2.2, 312.192.1-24.bt.2.3, 312.192.1-24.bt.2.4, 312.192.1-24.bt.2.5, 312.192.1-24.bt.2.6, 312.192.1-24.bt.2.7, 312.192.1-24.bt.2.8
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

$ 0 $ $=$ $ 2 x^{2} + 2 y^{2} + w^{2} $
$=$ $4 x^{2} - 2 y^{2} + z^{2} - 4 w^{2}$
 Copy content Toggle raw display

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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3^4}\cdot\frac{(z^{8}-12z^{6}w^{2}+45z^{4}w^{4}-54z^{2}w^{6}+81w^{8})^{3}}{w^{8}z^{4}(z^{2}-6w^{2})^{2}(z^{2}-3w^{2})^{4}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.g.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.k.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.l.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.y.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.bc.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bd.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bt.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.bkj.1 $24$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
24.384.17.nn.2 $24$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$