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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
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 $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot8$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1227

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&4\\16&11\end{bmatrix}$, $\begin{bmatrix}3&7\\16&17\end{bmatrix}$, $\begin{bmatrix}15&1\\8&9\end{bmatrix}$, $\begin{bmatrix}17&2\\16&21\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1089004
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.ct.1.1, 24.192.1-24.ct.1.2, 24.192.1-24.ct.1.3, 24.192.1-24.ct.1.4, 24.192.1-24.ct.1.5, 24.192.1-24.ct.1.6, 24.192.1-24.ct.1.7, 24.192.1-24.ct.1.8, 48.192.1-24.ct.1.1, 48.192.1-24.ct.1.2, 48.192.1-24.ct.1.3, 48.192.1-24.ct.1.4, 48.192.1-24.ct.1.5, 48.192.1-24.ct.1.6, 48.192.1-24.ct.1.7, 48.192.1-24.ct.1.8, 48.192.1-24.ct.1.9, 48.192.1-24.ct.1.10, 48.192.1-24.ct.1.11, 48.192.1-24.ct.1.12, 48.192.1-24.ct.1.13, 48.192.1-24.ct.1.14, 48.192.1-24.ct.1.15, 48.192.1-24.ct.1.16, 120.192.1-24.ct.1.1, 120.192.1-24.ct.1.2, 120.192.1-24.ct.1.3, 120.192.1-24.ct.1.4, 120.192.1-24.ct.1.5, 120.192.1-24.ct.1.6, 120.192.1-24.ct.1.7, 120.192.1-24.ct.1.8, 168.192.1-24.ct.1.1, 168.192.1-24.ct.1.2, 168.192.1-24.ct.1.3, 168.192.1-24.ct.1.4, 168.192.1-24.ct.1.5, 168.192.1-24.ct.1.6, 168.192.1-24.ct.1.7, 168.192.1-24.ct.1.8, 240.192.1-24.ct.1.1, 240.192.1-24.ct.1.2, 240.192.1-24.ct.1.3, 240.192.1-24.ct.1.4, 240.192.1-24.ct.1.5, 240.192.1-24.ct.1.6, 240.192.1-24.ct.1.7, 240.192.1-24.ct.1.8, 240.192.1-24.ct.1.9, 240.192.1-24.ct.1.10, 240.192.1-24.ct.1.11, 240.192.1-24.ct.1.12, 240.192.1-24.ct.1.13, 240.192.1-24.ct.1.14, 240.192.1-24.ct.1.15, 240.192.1-24.ct.1.16, 264.192.1-24.ct.1.1, 264.192.1-24.ct.1.2, 264.192.1-24.ct.1.3, 264.192.1-24.ct.1.4, 264.192.1-24.ct.1.5, 264.192.1-24.ct.1.6, 264.192.1-24.ct.1.7, 264.192.1-24.ct.1.8, 312.192.1-24.ct.1.1, 312.192.1-24.ct.1.2, 312.192.1-24.ct.1.3, 312.192.1-24.ct.1.4, 312.192.1-24.ct.1.5, 312.192.1-24.ct.1.6, 312.192.1-24.ct.1.7, 312.192.1-24.ct.1.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 3 x y - z w $
$=$ $3 x^{2} + 6 y^{2} + 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

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

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{(2z^{2}-4zw-w^{2})^{3}(2z^{2}+4zw-w^{2})^{3}(4z^{4}+12z^{2}w^{2}+w^{4})^{3}}{w^{4}z^{4}(2z^{2}-w^{2})^{8}}$

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
8.48.1.bc.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.be.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.be.2 $24$ $2$ $2$ $0$ $0$ 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
24.288.17.ckq.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.ro.1 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
48.192.5.gn.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.192.5.go.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.gx.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.he.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hf.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hf.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hg.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hg.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.hk.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hr.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ht.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.192.5.hu.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.9.ne.1 $48$ $2$ $2$ $9$ $3$ $1^{2}\cdot2^{3}$
48.192.9.ne.2 $48$ $2$ $2$ $9$ $3$ $1^{2}\cdot2^{3}$
48.192.9.nf.1 $48$ $2$ $2$ $9$ $3$ $1^{2}\cdot2^{3}$
48.192.9.nf.2 $48$ $2$ $2$ $9$ $3$ $1^{2}\cdot2^{3}$
240.192.5.bst.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bsv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.buc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bui.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvi.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bvv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bwb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bxa.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bxc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.cho.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cho.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.chp.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.chp.2 $240$ $2$ $2$ $9$ $?$ not computed