↑

Properties

Label 24.48.1.gd.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: $32$
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 $2^{4}$
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: 24.48.1.402

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&20\\0&7\end{bmatrix}$, $\begin{bmatrix}9&7\\22&19\end{bmatrix}$, $\begin{bmatrix}13&4\\22&23\end{bmatrix}$, $\begin{bmatrix}23&21\\8&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.gd.1.1, 24.96.1-24.gd.1.2, 120.96.1-24.gd.1.1, 120.96.1-24.gd.1.2, 168.96.1-24.gd.1.1, 168.96.1-24.gd.1.2, 264.96.1-24.gd.1.1, 264.96.1-24.gd.1.2, 312.96.1-24.gd.1.1, 312.96.1-24.gd.1.2
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ x^{2} + x y + x w + y^{2} - y w $
$=$ $x^{2} - 2 x y + 3 x z + x w + y^{2} - 3 y z - y w + 3 z^{2} - w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 3 x^{3} y + 5 x^{3} z - 3 x^{2} y^{2} + 9 x^{2} y z - 3 x^{2} z^{2} + 6 x y^{2} z - 9 x y z^{2} + \cdots + 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^4\cdot3^3\,\frac{(z-w)^{3}(3z^{2}-6zw-5w^{2})(12xz^{2}w^{4}-24xzw^{5}-4xw^{6}-12yz^{2}w^{4}+24yzw^{5}+4yw^{6}+9z^{7}-63z^{6}w+141z^{5}w^{2}-75z^{4}w^{3}-65z^{3}w^{4}+15z^{2}w^{5}+3zw^{6}+3w^{7})}{w^{8}(9xz^{3}-27xz^{2}w+15xzw^{2}+3xw^{3}-9yz^{3}+27yz^{2}w-15yzw^{2}-3yw^{3}+18z^{4}-54z^{3}w+24z^{2}w^{2}+18zw^{3}-10w^{4})}$

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
8.24.1.x.1 $8$ $2$ $2$ $1$ $0$ dimension zero
12.24.0.l.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.cf.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dj.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dx.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.ba.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.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
24.144.9.bnv.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.mr.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
120.240.17.xn.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.yyl.1 $120$ $6$ $6$ $17$ $?$ not computed