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Properties

Label 24.48.1.eu.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 $2^{4}$
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: 24.48.1.8

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&9\\8&17\end{bmatrix}$, $\begin{bmatrix}11&16\\4&15\end{bmatrix}$, $\begin{bmatrix}13&11\\0&23\end{bmatrix}$, $\begin{bmatrix}17&8\\8&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.eu.1.1, 24.96.1-24.eu.1.2, 24.96.1-24.eu.1.3, 24.96.1-24.eu.1.4, 120.96.1-24.eu.1.1, 120.96.1-24.eu.1.2, 120.96.1-24.eu.1.3, 120.96.1-24.eu.1.4, 168.96.1-24.eu.1.1, 168.96.1-24.eu.1.2, 168.96.1-24.eu.1.3, 168.96.1-24.eu.1.4, 264.96.1-24.eu.1.1, 264.96.1-24.eu.1.2, 264.96.1-24.eu.1.3, 264.96.1-24.eu.1.4, 312.96.1-24.eu.1.1, 312.96.1-24.eu.1.2, 312.96.1-24.eu.1.3, 312.96.1-24.eu.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 $ $=$ $ 2 x^{2} + 2 x y - 4 x z - y^{2} $
$=$ $x^{2} + x y - 2 x z + y^{2} - 3 y z + 3 z^{2} - 4 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 3 x^{2} y^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{4}{3}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 3^2\,\frac{z^{3}(3z^{2}-8w^{2})(216yz^{4}w^{2}-576yz^{2}w^{4}-512yw^{6}-27z^{7}-72z^{5}w^{2}+960z^{3}w^{4}-1024zw^{6})}{w^{8}(24yzw^{2}-9z^{4}+16w^{4})}$

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
4.24.0.c.1 $4$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.bc.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dw.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dx.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.m.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.do.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.dp.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.bhw.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.ks.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
120.240.17.vg.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.ujc.1 $120$ $6$ $6$ $17$ $?$ not computed