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Properties

Label 24.96.1.cx.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $48$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{4}\cdot4^{2}$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1840

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&16\\12&17\end{bmatrix}$, $\begin{bmatrix}7&9\\12&1\end{bmatrix}$, $\begin{bmatrix}11&10\\0&7\end{bmatrix}$, $\begin{bmatrix}17&21\\0&7\end{bmatrix}$, $\begin{bmatrix}17&22\\0&5\end{bmatrix}$, $\begin{bmatrix}19&20\\12&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_4^2:C_2^2\times D_6$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cx.2.1, 24.192.1-24.cx.2.2, 24.192.1-24.cx.2.3, 24.192.1-24.cx.2.4, 24.192.1-24.cx.2.5, 24.192.1-24.cx.2.6, 24.192.1-24.cx.2.7, 24.192.1-24.cx.2.8, 24.192.1-24.cx.2.9, 24.192.1-24.cx.2.10, 24.192.1-24.cx.2.11, 24.192.1-24.cx.2.12, 24.192.1-24.cx.2.13, 24.192.1-24.cx.2.14, 24.192.1-24.cx.2.15, 24.192.1-24.cx.2.16, 24.192.1-24.cx.2.17, 24.192.1-24.cx.2.18, 24.192.1-24.cx.2.19, 24.192.1-24.cx.2.20, 24.192.1-24.cx.2.21, 24.192.1-24.cx.2.22, 24.192.1-24.cx.2.23, 24.192.1-24.cx.2.24, 120.192.1-24.cx.2.1, 120.192.1-24.cx.2.2, 120.192.1-24.cx.2.3, 120.192.1-24.cx.2.4, 120.192.1-24.cx.2.5, 120.192.1-24.cx.2.6, 120.192.1-24.cx.2.7, 120.192.1-24.cx.2.8, 120.192.1-24.cx.2.9, 120.192.1-24.cx.2.10, 120.192.1-24.cx.2.11, 120.192.1-24.cx.2.12, 120.192.1-24.cx.2.13, 120.192.1-24.cx.2.14, 120.192.1-24.cx.2.15, 120.192.1-24.cx.2.16, 120.192.1-24.cx.2.17, 120.192.1-24.cx.2.18, 120.192.1-24.cx.2.19, 120.192.1-24.cx.2.20, 120.192.1-24.cx.2.21, 120.192.1-24.cx.2.22, 120.192.1-24.cx.2.23, 120.192.1-24.cx.2.24, 168.192.1-24.cx.2.1, 168.192.1-24.cx.2.2, 168.192.1-24.cx.2.3, 168.192.1-24.cx.2.4, 168.192.1-24.cx.2.5, 168.192.1-24.cx.2.6, 168.192.1-24.cx.2.7, 168.192.1-24.cx.2.8, 168.192.1-24.cx.2.9, 168.192.1-24.cx.2.10, 168.192.1-24.cx.2.11, 168.192.1-24.cx.2.12, 168.192.1-24.cx.2.13, 168.192.1-24.cx.2.14, 168.192.1-24.cx.2.15, 168.192.1-24.cx.2.16, 168.192.1-24.cx.2.17, 168.192.1-24.cx.2.18, 168.192.1-24.cx.2.19, 168.192.1-24.cx.2.20, 168.192.1-24.cx.2.21, 168.192.1-24.cx.2.22, 168.192.1-24.cx.2.23, 168.192.1-24.cx.2.24, 264.192.1-24.cx.2.1, 264.192.1-24.cx.2.2, 264.192.1-24.cx.2.3, 264.192.1-24.cx.2.4, 264.192.1-24.cx.2.5, 264.192.1-24.cx.2.6, 264.192.1-24.cx.2.7, 264.192.1-24.cx.2.8, 264.192.1-24.cx.2.9, 264.192.1-24.cx.2.10, 264.192.1-24.cx.2.11, 264.192.1-24.cx.2.12, 264.192.1-24.cx.2.13, 264.192.1-24.cx.2.14, 264.192.1-24.cx.2.15, 264.192.1-24.cx.2.16, 264.192.1-24.cx.2.17, 264.192.1-24.cx.2.18, 264.192.1-24.cx.2.19, 264.192.1-24.cx.2.20, 264.192.1-24.cx.2.21, 264.192.1-24.cx.2.22, 264.192.1-24.cx.2.23, 264.192.1-24.cx.2.24, 312.192.1-24.cx.2.1, 312.192.1-24.cx.2.2, 312.192.1-24.cx.2.3, 312.192.1-24.cx.2.4, 312.192.1-24.cx.2.5, 312.192.1-24.cx.2.6, 312.192.1-24.cx.2.7, 312.192.1-24.cx.2.8, 312.192.1-24.cx.2.9, 312.192.1-24.cx.2.10, 312.192.1-24.cx.2.11, 312.192.1-24.cx.2.12, 312.192.1-24.cx.2.13, 312.192.1-24.cx.2.14, 312.192.1-24.cx.2.15, 312.192.1-24.cx.2.16, 312.192.1-24.cx.2.17, 312.192.1-24.cx.2.18, 312.192.1-24.cx.2.19, 312.192.1-24.cx.2.20, 312.192.1-24.cx.2.21, 312.192.1-24.cx.2.22, 312.192.1-24.cx.2.23, 312.192.1-24.cx.2.24
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{4}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 48.2.a.a

Models

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

$ 0 $ $=$ $ 6 x^{2} - 6 y^{2} - w^{2} $
$=$ $12 x y + 6 y^{2} - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 2 x^{2} y^{2} + 4 x^{2} z^{2} - 12 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:

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

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{(z-w)^{3}(z+w)^{3}(1092y^{2}z^{16}+9552y^{2}z^{14}w^{2}+24528y^{2}z^{12}w^{4}+379056y^{2}z^{10}w^{6}+166872y^{2}z^{8}w^{8}+379056y^{2}z^{6}w^{10}+24528y^{2}z^{4}w^{12}+9552y^{2}z^{2}w^{14}+1092y^{2}w^{16}-61z^{18}-413z^{16}w^{2}-4116z^{14}w^{4}+40484z^{12}w^{6}+42538z^{10}w^{8}+48450z^{8}w^{10}+26780z^{6}w^{12}+9796z^{4}w^{14}+2187z^{2}w^{16}+243w^{18})}{w^{2}z^{2}(z^{2}+w^{2})^{4}(6y^{2}z^{10}+42y^{2}z^{8}w^{2}-156y^{2}z^{6}w^{4}+156y^{2}z^{4}w^{6}-42y^{2}z^{2}w^{8}-6y^{2}w^{10}+z^{12}+8z^{10}w^{2}+42z^{8}w^{4}+32z^{6}w^{6}+61z^{4}w^{8})}$

Modular covers

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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
12.48.1.h.1 $12$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bu.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.4 $24$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.192.5.dp.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ds.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dv.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.dy.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eb.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ed.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ee.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eg.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ei.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ej.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eo.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ep.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.9.jw.1 $24$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
24.192.9.kd.1 $24$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
24.192.9.ki.1 $24$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
24.192.9.kp.1 $24$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
24.288.9.m.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
72.288.9.j.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.bu.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.cg.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.sl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.sp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.sx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.tb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.to.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.tp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.tr.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ts.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ub.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.un.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ur.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.9.beq.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bet.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bfc.1 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.bff.1 $120$ $2$ $2$ $9$ $?$ not computed
168.192.5.sl.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.sp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.sx.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.tb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.to.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.tp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.tr.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ts.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ub.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.un.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ur.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.9.bef.1 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.bei.1 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.ber.1 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.beu.1 $168$ $2$ $2$ $9$ $?$ not computed
264.192.5.sl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.sp.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.sx.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.tb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.to.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.tp.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.tr.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ts.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ub.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uf.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.un.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ur.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.9.bdk.1 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.bdn.1 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.bdw.1 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.bdz.1 $264$ $2$ $2$ $9$ $?$ not computed
312.192.5.sl.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.sp.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.sx.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.tb.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.to.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.tp.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.tr.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ts.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ub.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uf.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.un.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ur.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.9.beq.2 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.bet.2 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.bfc.1 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.bff.1 $312$ $2$ $2$ $9$ $?$ not computed