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Properties

Label 24.192.7.ct.1
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: 24.192.7.106

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\0&23\end{bmatrix}$, $\begin{bmatrix}5&16\\0&17\end{bmatrix}$, $\begin{bmatrix}11&6\\12&19\end{bmatrix}$, $\begin{bmatrix}11&16\\0&13\end{bmatrix}$, $\begin{bmatrix}13&18\\0&13\end{bmatrix}$, $\begin{bmatrix}19&20\\12&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{12}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 24.384.7-24.ct.1.1, 24.384.7-24.ct.1.2, 24.384.7-24.ct.1.3, 24.384.7-24.ct.1.4, 24.384.7-24.ct.1.5, 24.384.7-24.ct.1.6, 24.384.7-24.ct.1.7, 24.384.7-24.ct.1.8, 24.384.7-24.ct.1.9, 24.384.7-24.ct.1.10, 24.384.7-24.ct.1.11, 24.384.7-24.ct.1.12, 24.384.7-24.ct.1.13, 24.384.7-24.ct.1.14, 24.384.7-24.ct.1.15, 24.384.7-24.ct.1.16, 24.384.7-24.ct.1.17, 24.384.7-24.ct.1.18, 24.384.7-24.ct.1.19, 24.384.7-24.ct.1.20, 24.384.7-24.ct.1.21, 24.384.7-24.ct.1.22, 24.384.7-24.ct.1.23, 24.384.7-24.ct.1.24, 24.384.7-24.ct.1.25, 24.384.7-24.ct.1.26, 24.384.7-24.ct.1.27, 24.384.7-24.ct.1.28, 24.384.7-24.ct.1.29, 24.384.7-24.ct.1.30, 24.384.7-24.ct.1.31, 24.384.7-24.ct.1.32, 120.384.7-24.ct.1.1, 120.384.7-24.ct.1.2, 120.384.7-24.ct.1.3, 120.384.7-24.ct.1.4, 120.384.7-24.ct.1.5, 120.384.7-24.ct.1.6, 120.384.7-24.ct.1.7, 120.384.7-24.ct.1.8, 120.384.7-24.ct.1.9, 120.384.7-24.ct.1.10, 120.384.7-24.ct.1.11, 120.384.7-24.ct.1.12, 120.384.7-24.ct.1.13, 120.384.7-24.ct.1.14, 120.384.7-24.ct.1.15, 120.384.7-24.ct.1.16, 120.384.7-24.ct.1.17, 120.384.7-24.ct.1.18, 120.384.7-24.ct.1.19, 120.384.7-24.ct.1.20, 120.384.7-24.ct.1.21, 120.384.7-24.ct.1.22, 120.384.7-24.ct.1.23, 120.384.7-24.ct.1.24, 120.384.7-24.ct.1.25, 120.384.7-24.ct.1.26, 120.384.7-24.ct.1.27, 120.384.7-24.ct.1.28, 120.384.7-24.ct.1.29, 120.384.7-24.ct.1.30, 120.384.7-24.ct.1.31, 120.384.7-24.ct.1.32, 168.384.7-24.ct.1.1, 168.384.7-24.ct.1.2, 168.384.7-24.ct.1.3, 168.384.7-24.ct.1.4, 168.384.7-24.ct.1.5, 168.384.7-24.ct.1.6, 168.384.7-24.ct.1.7, 168.384.7-24.ct.1.8, 168.384.7-24.ct.1.9, 168.384.7-24.ct.1.10, 168.384.7-24.ct.1.11, 168.384.7-24.ct.1.12, 168.384.7-24.ct.1.13, 168.384.7-24.ct.1.14, 168.384.7-24.ct.1.15, 168.384.7-24.ct.1.16, 168.384.7-24.ct.1.17, 168.384.7-24.ct.1.18, 168.384.7-24.ct.1.19, 168.384.7-24.ct.1.20, 168.384.7-24.ct.1.21, 168.384.7-24.ct.1.22, 168.384.7-24.ct.1.23, 168.384.7-24.ct.1.24, 168.384.7-24.ct.1.25, 168.384.7-24.ct.1.26, 168.384.7-24.ct.1.27, 168.384.7-24.ct.1.28, 168.384.7-24.ct.1.29, 168.384.7-24.ct.1.30, 168.384.7-24.ct.1.31, 168.384.7-24.ct.1.32, 264.384.7-24.ct.1.1, 264.384.7-24.ct.1.2, 264.384.7-24.ct.1.3, 264.384.7-24.ct.1.4, 264.384.7-24.ct.1.5, 264.384.7-24.ct.1.6, 264.384.7-24.ct.1.7, 264.384.7-24.ct.1.8, 264.384.7-24.ct.1.9, 264.384.7-24.ct.1.10, 264.384.7-24.ct.1.11, 264.384.7-24.ct.1.12, 264.384.7-24.ct.1.13, 264.384.7-24.ct.1.14, 264.384.7-24.ct.1.15, 264.384.7-24.ct.1.16, 264.384.7-24.ct.1.17, 264.384.7-24.ct.1.18, 264.384.7-24.ct.1.19, 264.384.7-24.ct.1.20, 264.384.7-24.ct.1.21, 264.384.7-24.ct.1.22, 264.384.7-24.ct.1.23, 264.384.7-24.ct.1.24, 264.384.7-24.ct.1.25, 264.384.7-24.ct.1.26, 264.384.7-24.ct.1.27, 264.384.7-24.ct.1.28, 264.384.7-24.ct.1.29, 264.384.7-24.ct.1.30, 264.384.7-24.ct.1.31, 264.384.7-24.ct.1.32, 312.384.7-24.ct.1.1, 312.384.7-24.ct.1.2, 312.384.7-24.ct.1.3, 312.384.7-24.ct.1.4, 312.384.7-24.ct.1.5, 312.384.7-24.ct.1.6, 312.384.7-24.ct.1.7, 312.384.7-24.ct.1.8, 312.384.7-24.ct.1.9, 312.384.7-24.ct.1.10, 312.384.7-24.ct.1.11, 312.384.7-24.ct.1.12, 312.384.7-24.ct.1.13, 312.384.7-24.ct.1.14, 312.384.7-24.ct.1.15, 312.384.7-24.ct.1.16, 312.384.7-24.ct.1.17, 312.384.7-24.ct.1.18, 312.384.7-24.ct.1.19, 312.384.7-24.ct.1.20, 312.384.7-24.ct.1.21, 312.384.7-24.ct.1.22, 312.384.7-24.ct.1.23, 312.384.7-24.ct.1.24, 312.384.7-24.ct.1.25, 312.384.7-24.ct.1.26, 312.384.7-24.ct.1.27, 312.384.7-24.ct.1.28, 312.384.7-24.ct.1.29, 312.384.7-24.ct.1.30, 312.384.7-24.ct.1.31, 312.384.7-24.ct.1.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
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 $ $=$ $ y v - w t $
$=$ $x u - y z$
$=$ $3 y w - u v$
$=$ $3 y^{2} - t u$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 2 x^{6} y^{2} + 3 x^{6} z^{2} - 4 x^{4} y^{4} + 18 x^{4} y^{2} z^{2} - 18 x^{4} z^{4} + \cdots + 54 y^{2} z^{6} $
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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:

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

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.bn.1 :

$\displaystyle X$ $=$ $\displaystyle 2y$
$\displaystyle Y$ $=$ $\displaystyle -x+w$
$\displaystyle Z$ $=$ $\displaystyle x+w$

Equation of the image curve:

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

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
24.48.0.t.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.96.3.bn.1 $24$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$
24.96.3.bo.1 $24$ $2$ $2$ $3$ $0$ $1^{2}\cdot2$
24.96.3.ch.1 $24$ $2$ $2$ $3$ $0$ $2^{2}$

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.cm.1 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.cm.2 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.co.3 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.co.4 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.dc.1 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.dc.2 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.de.3 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.13.de.4 $24$ $2$ $2$ $13$ $0$ $2^{3}$
24.384.17.dg.1 $24$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
24.384.17.di.1 $24$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
24.384.17.du.1 $24$ $2$ $2$ $17$ $1$ $1^{6}\cdot2^{2}$
24.384.17.dv.1 $24$ $2$ $2$ $17$ $0$ $1^{6}\cdot2^{2}$
24.384.17.hl.1 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
24.384.17.hm.1 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
24.384.17.hn.1 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
24.384.17.ho.1 $24$ $2$ $2$ $17$ $2$ $1^{6}\cdot2^{2}$
24.384.17.pa.1 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pa.2 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pa.3 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pa.4 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pc.1 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pc.2 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pc.3 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.384.17.pc.4 $24$ $2$ $2$ $17$ $0$ $2^{3}\cdot4$
24.576.29.cf.2 $24$ $3$ $3$ $29$ $0$ $1^{10}\cdot2^{6}$
120.384.13.beq.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.beq.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bes.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bes.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bfo.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bfo.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bfq.2 $120$ $2$ $2$ $13$ $?$ not computed
120.384.13.bfq.4 $120$ $2$ $2$ $13$ $?$ not computed
120.384.17.dpe.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpf.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpg.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dph.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpv.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpw.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.dpx.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eje.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eje.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eje.3 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eje.4 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eji.1 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eji.2 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eji.3 $120$ $2$ $2$ $17$ $?$ not computed
120.384.17.eji.4 $120$ $2$ $2$ $17$ $?$ not computed
168.384.13.bdk.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bdk.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bdm.2 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bdm.3 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bei.3 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bei.4 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bek.1 $168$ $2$ $2$ $13$ $?$ not computed
168.384.13.bek.2 $168$ $2$ $2$ $13$ $?$ not computed
168.384.17.dpe.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpf.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dph.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpv.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpw.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.dpx.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eje.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eje.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eje.3 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eje.4 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eji.1 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eji.2 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eji.3 $168$ $2$ $2$ $17$ $?$ not computed
168.384.17.eji.4 $168$ $2$ $2$ $17$ $?$ not computed
264.384.13.bdk.1 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bdk.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bdm.3 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bdm.4 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bei.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bei.3 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bek.1 $264$ $2$ $2$ $13$ $?$ not computed
264.384.13.bek.2 $264$ $2$ $2$ $13$ $?$ not computed
264.384.17.dpe.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpf.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpg.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dph.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpu.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpv.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpw.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.dpx.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eje.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eje.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eje.3 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eje.4 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eji.1 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eji.2 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eji.3 $264$ $2$ $2$ $17$ $?$ not computed
264.384.17.eji.4 $264$ $2$ $2$ $17$ $?$ not computed
312.384.13.beq.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.beq.3 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bes.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bes.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bfo.1 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bfo.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bfq.2 $312$ $2$ $2$ $13$ $?$ not computed
312.384.13.bfq.4 $312$ $2$ $2$ $13$ $?$ not computed
312.384.17.dpe.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpf.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpg.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dph.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpu.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpv.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpw.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.dpx.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eje.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eje.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eje.5 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eje.6 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eji.1 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eji.2 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eji.5 $312$ $2$ $2$ $17$ $?$ not computed
312.384.17.eji.6 $312$ $2$ $2$ $17$ $?$ not computed