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Properties

Label 24.192.7.eh.2
Level $24$
Index $192$
Genus $7$
Analytic rank $0$
Cusps $20$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $288$
Index: $192$ $\PSL_2$-index:$192$
Genus: $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$
Cusps: $20$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot6^{4}\cdot8^{4}\cdot12^{2}\cdot24^{4}$ Cusp orbits $2^{6}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24AI7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.7.150

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&22\\0&13\end{bmatrix}$, $\begin{bmatrix}7&3\\0&11\end{bmatrix}$, $\begin{bmatrix}11&7\\0&1\end{bmatrix}$, $\begin{bmatrix}13&3\\0&11\end{bmatrix}$, $\begin{bmatrix}17&18\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $D_{12}:C_2^4$
Contains $-I$: yes
Quadratic refinements: 24.384.7-24.eh.2.1, 24.384.7-24.eh.2.2, 24.384.7-24.eh.2.3, 24.384.7-24.eh.2.4, 24.384.7-24.eh.2.5, 24.384.7-24.eh.2.6, 24.384.7-24.eh.2.7, 24.384.7-24.eh.2.8, 24.384.7-24.eh.2.9, 24.384.7-24.eh.2.10, 24.384.7-24.eh.2.11, 24.384.7-24.eh.2.12, 24.384.7-24.eh.2.13, 24.384.7-24.eh.2.14, 24.384.7-24.eh.2.15, 24.384.7-24.eh.2.16, 48.384.7-24.eh.2.1, 48.384.7-24.eh.2.2, 48.384.7-24.eh.2.3, 48.384.7-24.eh.2.4, 48.384.7-24.eh.2.5, 48.384.7-24.eh.2.6, 48.384.7-24.eh.2.7, 48.384.7-24.eh.2.8, 48.384.7-24.eh.2.9, 48.384.7-24.eh.2.10, 48.384.7-24.eh.2.11, 48.384.7-24.eh.2.12, 48.384.7-24.eh.2.13, 48.384.7-24.eh.2.14, 48.384.7-24.eh.2.15, 48.384.7-24.eh.2.16, 120.384.7-24.eh.2.1, 120.384.7-24.eh.2.2, 120.384.7-24.eh.2.3, 120.384.7-24.eh.2.4, 120.384.7-24.eh.2.5, 120.384.7-24.eh.2.6, 120.384.7-24.eh.2.7, 120.384.7-24.eh.2.8, 120.384.7-24.eh.2.9, 120.384.7-24.eh.2.10, 120.384.7-24.eh.2.11, 120.384.7-24.eh.2.12, 120.384.7-24.eh.2.13, 120.384.7-24.eh.2.14, 120.384.7-24.eh.2.15, 120.384.7-24.eh.2.16, 168.384.7-24.eh.2.1, 168.384.7-24.eh.2.2, 168.384.7-24.eh.2.3, 168.384.7-24.eh.2.4, 168.384.7-24.eh.2.5, 168.384.7-24.eh.2.6, 168.384.7-24.eh.2.7, 168.384.7-24.eh.2.8, 168.384.7-24.eh.2.9, 168.384.7-24.eh.2.10, 168.384.7-24.eh.2.11, 168.384.7-24.eh.2.12, 168.384.7-24.eh.2.13, 168.384.7-24.eh.2.14, 168.384.7-24.eh.2.15, 168.384.7-24.eh.2.16, 240.384.7-24.eh.2.1, 240.384.7-24.eh.2.2, 240.384.7-24.eh.2.3, 240.384.7-24.eh.2.4, 240.384.7-24.eh.2.5, 240.384.7-24.eh.2.6, 240.384.7-24.eh.2.7, 240.384.7-24.eh.2.8, 240.384.7-24.eh.2.9, 240.384.7-24.eh.2.10, 240.384.7-24.eh.2.11, 240.384.7-24.eh.2.12, 240.384.7-24.eh.2.13, 240.384.7-24.eh.2.14, 240.384.7-24.eh.2.15, 240.384.7-24.eh.2.16, 264.384.7-24.eh.2.1, 264.384.7-24.eh.2.2, 264.384.7-24.eh.2.3, 264.384.7-24.eh.2.4, 264.384.7-24.eh.2.5, 264.384.7-24.eh.2.6, 264.384.7-24.eh.2.7, 264.384.7-24.eh.2.8, 264.384.7-24.eh.2.9, 264.384.7-24.eh.2.10, 264.384.7-24.eh.2.11, 264.384.7-24.eh.2.12, 264.384.7-24.eh.2.13, 264.384.7-24.eh.2.14, 264.384.7-24.eh.2.15, 264.384.7-24.eh.2.16, 312.384.7-24.eh.2.1, 312.384.7-24.eh.2.2, 312.384.7-24.eh.2.3, 312.384.7-24.eh.2.4, 312.384.7-24.eh.2.5, 312.384.7-24.eh.2.6, 312.384.7-24.eh.2.7, 312.384.7-24.eh.2.8, 312.384.7-24.eh.2.9, 312.384.7-24.eh.2.10, 312.384.7-24.eh.2.11, 312.384.7-24.eh.2.12, 312.384.7-24.eh.2.13, 312.384.7-24.eh.2.14, 312.384.7-24.eh.2.15, 312.384.7-24.eh.2.16
Cyclic 24-isogeny field degree: $1$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{30}\cdot3^{11}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2^{2}$
Newforms: 24.2.a.a, 72.2.a.a, 96.2.d.a, 144.2.a.b, 288.2.d.b

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ z^{2} - u^{2} - u v - v^{2} $
$=$ $z w - z t - w u - t v$
$=$ $z w - w u - w v + t u$
$=$ $x t + y w + z u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{8} - 2 x^{7} y - 5 x^{6} y^{2} - x^{6} z^{2} - 2 x^{5} y^{3} + 6 x^{5} y z^{2} + 13 x^{4} y^{4} + \cdots + 8 y^{6} z^{2} $
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Rational points

This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.gf.2 :

$\displaystyle X$ $=$ $\displaystyle -x+w$
$\displaystyle Y$ $=$ $\displaystyle x+w$
$\displaystyle Z$ $=$ $\displaystyle -u$

Equation of the image curve:

$0$ $=$ $ X^{3}Y-XY^{3}-X^{2}Z^{2}-Y^{2}Z^{2}-2Z^{4} $

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0.bi.2 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.96.3.ez.1 $24$ $2$ $2$ $3$ $0$ $2^{2}$
24.96.3.gf.2 $24$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$
24.96.3.gg.2 $24$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.384.13.fk.2 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fk.4 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fm.1 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fm.3 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fo.2 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fo.4 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fq.1 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.fq.3 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.576.29.lj.1 $24$ $3$ $3$ $29$ $0$ $1^{10}\cdot2^{6}$
48.384.17.vv.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.wb.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.wl.1 $48$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
48.384.17.wr.2 $48$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
48.384.17.bdz.3 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bdz.4 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bdz.7 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bdz.8 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.beb.1 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.beb.2 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.beb.5 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.beb.6 $48$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
48.384.17.bkp.1 $48$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
48.384.17.bkv.2 $48$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
48.384.17.blf.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
48.384.17.bll.2 $48$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
120.384.13.blo.1 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.blo.3 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.blq.1 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.blq.3 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bls.1 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bls.3 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.blu.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.blu.4 $120$ $2$ $2$ $13$ $?$ not computed
168.384.13.bki.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bki.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bkk.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bkk.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bkm.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bkm.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bko.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bko.4 $168$ $2$ $2$ $13$ $?$ not computed
240.384.17.gvf.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.gvp.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.gvv.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.gwf.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hid.3 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hid.4 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hid.7 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hid.8 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hif.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hif.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hif.5 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.hif.6 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.ibv.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.icf.1 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.icl.2 $240$ $2$ $2$ $17$ $?$ not computed
240.384.17.icv.2 $240$ $2$ $2$ $17$ $?$ not computed
264.384.13.bki.1 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bki.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bkk.1 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bkk.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bkm.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bkm.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bko.1 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bko.3 $264$ $2$ $2$ $13$ $?$ not computed
312.384.13.blo.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.blo.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.blq.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.blq.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bls.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bls.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.blu.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.blu.4 $312$ $2$ $2$ $13$ $?$ not computed