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Properties

Label 24.48.1.ie.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: $12$ Newform level: $576$
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 $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ 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: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.507

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\6&7\end{bmatrix}$, $\begin{bmatrix}11&8\\0&7\end{bmatrix}$, $\begin{bmatrix}11&23\\0&23\end{bmatrix}$, $\begin{bmatrix}17&3\\18&1\end{bmatrix}$, $\begin{bmatrix}19&23\\0&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.ie.1.1, 24.96.1-24.ie.1.2, 24.96.1-24.ie.1.3, 24.96.1-24.ie.1.4, 24.96.1-24.ie.1.5, 24.96.1-24.ie.1.6, 24.96.1-24.ie.1.7, 24.96.1-24.ie.1.8, 24.96.1-24.ie.1.9, 24.96.1-24.ie.1.10, 24.96.1-24.ie.1.11, 24.96.1-24.ie.1.12, 120.96.1-24.ie.1.1, 120.96.1-24.ie.1.2, 120.96.1-24.ie.1.3, 120.96.1-24.ie.1.4, 120.96.1-24.ie.1.5, 120.96.1-24.ie.1.6, 120.96.1-24.ie.1.7, 120.96.1-24.ie.1.8, 120.96.1-24.ie.1.9, 120.96.1-24.ie.1.10, 120.96.1-24.ie.1.11, 120.96.1-24.ie.1.12, 168.96.1-24.ie.1.1, 168.96.1-24.ie.1.2, 168.96.1-24.ie.1.3, 168.96.1-24.ie.1.4, 168.96.1-24.ie.1.5, 168.96.1-24.ie.1.6, 168.96.1-24.ie.1.7, 168.96.1-24.ie.1.8, 168.96.1-24.ie.1.9, 168.96.1-24.ie.1.10, 168.96.1-24.ie.1.11, 168.96.1-24.ie.1.12, 264.96.1-24.ie.1.1, 264.96.1-24.ie.1.2, 264.96.1-24.ie.1.3, 264.96.1-24.ie.1.4, 264.96.1-24.ie.1.5, 264.96.1-24.ie.1.6, 264.96.1-24.ie.1.7, 264.96.1-24.ie.1.8, 264.96.1-24.ie.1.9, 264.96.1-24.ie.1.10, 264.96.1-24.ie.1.11, 264.96.1-24.ie.1.12, 312.96.1-24.ie.1.1, 312.96.1-24.ie.1.2, 312.96.1-24.ie.1.3, 312.96.1-24.ie.1.4, 312.96.1-24.ie.1.5, 312.96.1-24.ie.1.6, 312.96.1-24.ie.1.7, 312.96.1-24.ie.1.8, 312.96.1-24.ie.1.9, 312.96.1-24.ie.1.10, 312.96.1-24.ie.1.11, 312.96.1-24.ie.1.12
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

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

$ 0 $ $=$ $ x^{2} - y z $
$=$ $30 x^{2} + 6 y^{2} + 30 y z + 54 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} + 10 x^{2} z^{2} - 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}w$
$\displaystyle Z$ $=$ $\displaystyle 3z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{17390370816yz^{11}+3901685760yz^{9}w^{2}+304570368yz^{7}w^{4}+9234432yz^{5}w^{6}+75456yz^{3}w^{8}+144yzw^{10}+17199267840z^{12}+3551330304z^{10}w^{2}+223948800z^{8}w^{4}+3096576z^{6}w^{6}-101376z^{4}w^{8}-1008z^{2}w^{10}-w^{12}}{w^{2}z^{6}(23328yz^{3}+108yzw^{2}+23328z^{4}-378z^{2}w^{2}-w^{4})}$

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
12.24.0.f.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.m.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.24.0.ca.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.er.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
24.96.3.fi.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fi.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fj.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.fj.2 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.fv.1 $24$ $3$ $3$ $5$ $0$ $1^{4}$
72.144.5.y.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.ca.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.cb.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.3.lo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.lo.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.lp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.lp.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.bpq.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkeu.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.3.je.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.je.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.jf.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.jf.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.3.je.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.je.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.jf.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.jf.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.3.lo.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.lo.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.lp.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.lp.2 $312$ $2$ $2$ $3$ $?$ not computed