↑

Properties

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

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
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: 24.48.1.386

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}9&14\\20&17\end{bmatrix}$, $\begin{bmatrix}19&6\\12&11\end{bmatrix}$, $\begin{bmatrix}23&5\\0&1\end{bmatrix}$, $\begin{bmatrix}23&5\\12&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.ea.1.1, 24.96.1-24.ea.1.2, 24.96.1-24.ea.1.3, 24.96.1-24.ea.1.4, 48.96.1-24.ea.1.1, 48.96.1-24.ea.1.2, 48.96.1-24.ea.1.3, 48.96.1-24.ea.1.4, 120.96.1-24.ea.1.1, 120.96.1-24.ea.1.2, 120.96.1-24.ea.1.3, 120.96.1-24.ea.1.4, 168.96.1-24.ea.1.1, 168.96.1-24.ea.1.2, 168.96.1-24.ea.1.3, 168.96.1-24.ea.1.4, 240.96.1-24.ea.1.1, 240.96.1-24.ea.1.2, 240.96.1-24.ea.1.3, 240.96.1-24.ea.1.4, 264.96.1-24.ea.1.1, 264.96.1-24.ea.1.2, 264.96.1-24.ea.1.3, 264.96.1-24.ea.1.4, 312.96.1-24.ea.1.1, 312.96.1-24.ea.1.2, 312.96.1-24.ea.1.3, 312.96.1-24.ea.1.4
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

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 $ $=$ $ x^{2} + x z - 6 y^{2} + z^{2} $
$=$ $7 x^{2} + 4 x z + 4 z^{2} - w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 20 x^{2} y^{2} + 3 x^{2} z^{2} + 196 y^{4} - 84 y^{2} z^{2} + 9 z^{4} $
 Copy content Toggle raw display

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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^2}\cdot\frac{30195296640xz^{11}-63407183616xz^{9}w^{2}+28300176576xz^{7}w^{4}+385850304xz^{5}w^{6}-642565224xz^{3}w^{8}+121010400xzw^{10}+9922751424z^{12}-51334523712z^{10}w^{2}+59671379376z^{8}w^{4}-25404182496z^{6}w^{6}+4834279044z^{4}w^{8}-230351940z^{2}w^{10}-2100875w^{12}}{31065120xz^{11}+7764120xz^{9}w^{2}-12674340xz^{7}w^{4}+1382976xz^{5}w^{6}+941192xz^{3}w^{8}-67228xzw^{10}+10208592z^{12}+24629832z^{10}w^{2}-5387823z^{8}w^{4}-2916186z^{6}w^{6}+929187z^{4}w^{8}-4802z^{2}w^{10}-16807w^{12}}$

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
8.24.1.n.1 $8$ $2$ $2$ $1$ $0$ dimension zero
12.24.0.e.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.u.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.es.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.et.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.bg.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.bh.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.za.1 $24$ $3$ $3$ $9$ $0$ $1^{8}$
24.192.9.jc.1 $24$ $4$ $4$ $9$ $3$ $1^{8}$
48.96.5.gb.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.gb.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.gc.1 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
48.96.5.gc.2 $48$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
120.240.17.pc.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.rmu.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.5.qv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.qv.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.qw.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.qw.2 $240$ $2$ $2$ $5$ $?$ not computed