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Properties

Label 24.96.1.cd.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: $8$ Newform level: $288$
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 $2^{6}\cdot4$
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: 24.96.1.489

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&22\\8&5\end{bmatrix}$, $\begin{bmatrix}7&20\\20&5\end{bmatrix}$, $\begin{bmatrix}17&0\\12&23\end{bmatrix}$, $\begin{bmatrix}23&20\\0&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1089047
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cd.2.1, 24.192.1-24.cd.2.2, 24.192.1-24.cd.2.3, 24.192.1-24.cd.2.4, 24.192.1-24.cd.2.5, 24.192.1-24.cd.2.6, 24.192.1-24.cd.2.7, 24.192.1-24.cd.2.8, 48.192.1-24.cd.2.1, 48.192.1-24.cd.2.2, 48.192.1-24.cd.2.3, 48.192.1-24.cd.2.4, 48.192.1-24.cd.2.5, 48.192.1-24.cd.2.6, 48.192.1-24.cd.2.7, 48.192.1-24.cd.2.8, 48.192.1-24.cd.2.9, 48.192.1-24.cd.2.10, 48.192.1-24.cd.2.11, 48.192.1-24.cd.2.12, 120.192.1-24.cd.2.1, 120.192.1-24.cd.2.2, 120.192.1-24.cd.2.3, 120.192.1-24.cd.2.4, 120.192.1-24.cd.2.5, 120.192.1-24.cd.2.6, 120.192.1-24.cd.2.7, 120.192.1-24.cd.2.8, 168.192.1-24.cd.2.1, 168.192.1-24.cd.2.2, 168.192.1-24.cd.2.3, 168.192.1-24.cd.2.4, 168.192.1-24.cd.2.5, 168.192.1-24.cd.2.6, 168.192.1-24.cd.2.7, 168.192.1-24.cd.2.8, 240.192.1-24.cd.2.1, 240.192.1-24.cd.2.2, 240.192.1-24.cd.2.3, 240.192.1-24.cd.2.4, 240.192.1-24.cd.2.5, 240.192.1-24.cd.2.6, 240.192.1-24.cd.2.7, 240.192.1-24.cd.2.8, 240.192.1-24.cd.2.9, 240.192.1-24.cd.2.10, 240.192.1-24.cd.2.11, 240.192.1-24.cd.2.12, 264.192.1-24.cd.2.1, 264.192.1-24.cd.2.2, 264.192.1-24.cd.2.3, 264.192.1-24.cd.2.4, 264.192.1-24.cd.2.5, 264.192.1-24.cd.2.6, 264.192.1-24.cd.2.7, 264.192.1-24.cd.2.8, 312.192.1-24.cd.2.1, 312.192.1-24.cd.2.2, 312.192.1-24.cd.2.3, 312.192.1-24.cd.2.4, 312.192.1-24.cd.2.5, 312.192.1-24.cd.2.6, 312.192.1-24.cd.2.7, 312.192.1-24.cd.2.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

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

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

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}w$
$\displaystyle Z$ $=$ $\displaystyle y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^2}\cdot\frac{577506008273040xz^{23}+1256929747798752xz^{21}w^{2}+1284395398125552xz^{19}w^{4}+810298193070816xz^{17}w^{6}+349020510694560xz^{15}w^{8}+107201322818496xz^{13}w^{10}+23801518812384xz^{11}w^{12}+3785206396608xz^{9}w^{14}+416636272080xz^{7}w^{16}+29496753504xz^{5}w^{18}+1152466992xz^{3}w^{20}+16882272xzw^{22}-408358414625841z^{24}-956843283914568z^{22}w^{2}-1050663642449058z^{20}w^{4}-712935854554968z^{18}w^{6}-331535810847015z^{16}w^{8}-110697022043856z^{14}w^{10}-27013159359036z^{12}w^{12}-4803935821488z^{10}w^{14}-608054013999z^{8}w^{16}-51983518824z^{6}w^{18}-2703477762z^{4}w^{20}-67944312z^{2}w^{22}-389017w^{24}}{w^{4}(3z^{2}+w^{2})^{2}(23319876024xz^{15}+26101660320xz^{13}w^{2}+11524027464xz^{11}w^{4}+2545575984xz^{9}w^{6}+294284808xz^{7}w^{8}+16937088xz^{5}w^{10}+412152xz^{3}w^{12}+2736xzw^{14}-16489642473z^{16}-21204934758z^{14}w^{2}-10995805323z^{12}w^{4}-2939941656z^{10}w^{6}-429684723z^{8}w^{8}-33475710z^{6}w^{10}-1247221z^{4}w^{12}-17028z^{2}w^{14}-36w^{16})}$

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
$X_{\pm1}(2,8)$ $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.i.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.j.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bb.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.be.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bf.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bu.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
24.288.17.blf.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.od.1 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
48.192.5.bq.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.cb.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.dj.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dk.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dn.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.do.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dr.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ds.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dv.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dw.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ef.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ek.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
240.192.5.sx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.tf.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.uv.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.uw.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vl.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vm.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.xf.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.xn.2 $240$ $2$ $2$ $5$ $?$ not computed