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Properties

Label 24.48.1.eb.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{2}\cdot4$
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: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.206

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&1\\4&5\end{bmatrix}$, $\begin{bmatrix}7&14\\0&11\end{bmatrix}$, $\begin{bmatrix}11&4\\0&11\end{bmatrix}$, $\begin{bmatrix}19&18\\20&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.eb.1.1, 24.96.1-24.eb.1.2, 24.96.1-24.eb.1.3, 24.96.1-24.eb.1.4, 48.96.1-24.eb.1.1, 48.96.1-24.eb.1.2, 48.96.1-24.eb.1.3, 48.96.1-24.eb.1.4, 120.96.1-24.eb.1.1, 120.96.1-24.eb.1.2, 120.96.1-24.eb.1.3, 120.96.1-24.eb.1.4, 168.96.1-24.eb.1.1, 168.96.1-24.eb.1.2, 168.96.1-24.eb.1.3, 168.96.1-24.eb.1.4, 240.96.1-24.eb.1.1, 240.96.1-24.eb.1.2, 240.96.1-24.eb.1.3, 240.96.1-24.eb.1.4, 264.96.1-24.eb.1.1, 264.96.1-24.eb.1.2, 264.96.1-24.eb.1.3, 264.96.1-24.eb.1.4, 312.96.1-24.eb.1.1, 312.96.1-24.eb.1.2, 312.96.1-24.eb.1.3, 312.96.1-24.eb.1.4
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

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 $ $=$ $ x^{2} + y z $
$=$ $3 y^{2} + 3 z^{2} - 8 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 6 y^{2} z^{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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

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

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
8.24.0.t.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.u.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.do.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dp.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.m.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.dg.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.dh.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
24.144.9.zb.1 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.192.9.jd.1 $24$ $4$ $4$ $9$ $3$ $1^{8}$
48.96.3.gz.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.gz.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ha.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ha.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
120.240.17.pd.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.rmv.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.3.tx.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tx.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ty.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ty.2 $240$ $2$ $2$ $3$ $?$ not computed