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Properties

Label 24.144.5.gb.1
Level $24$
Index $144$
Genus $5$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $144$ $\PSL_2$-index:$144$
Genus: $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $6^{8}\cdot12^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12B5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.5.583

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&9\\18&11\end{bmatrix}$, $\begin{bmatrix}11&9\\0&5\end{bmatrix}$, $\begin{bmatrix}11&12\\18&17\end{bmatrix}$, $\begin{bmatrix}19&16\\12&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.gb.1.1, 24.288.5-24.gb.1.2, 24.288.5-24.gb.1.3, 24.288.5-24.gb.1.4, 120.288.5-24.gb.1.1, 120.288.5-24.gb.1.2, 120.288.5-24.gb.1.3, 120.288.5-24.gb.1.4, 168.288.5-24.gb.1.1, 168.288.5-24.gb.1.2, 168.288.5-24.gb.1.3, 168.288.5-24.gb.1.4, 264.288.5-24.gb.1.1, 264.288.5-24.gb.1.2, 264.288.5-24.gb.1.3, 264.288.5-24.gb.1.4, 312.288.5-24.gb.1.1, 312.288.5-24.gb.1.2, 312.288.5-24.gb.1.3, 312.288.5-24.gb.1.4
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{24}\cdot3^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 48.2.a.a, 144.2.a.b$^{2}$, 576.2.a.e, 576.2.a.f

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

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

$ 0 $ $=$ $ - 9 x^{4} y^{4} - 36 x^{3} y^{5} + 18 x^{3} y^{3} z^{2} - 252 x^{2} y^{6} + 156 x^{2} y^{4} z^{2} + \cdots + 2 z^{8} $
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Rational points

This modular curve has no $\Q_p$ points for $p=37$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}t$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{5039360yw^{17}-27713408yw^{15}t^{2}+17636224yw^{13}t^{4}+43669376yw^{11}t^{6}-65295776yw^{9}t^{8}+32542464yw^{7}t^{10}-5830152yw^{5}t^{12}-194400yw^{3}t^{14}+124508ywt^{16}-256z^{18}-1152z^{14}t^{4}+768z^{12}t^{6}-1584z^{10}t^{8}+2592z^{8}t^{10}-4608z^{6}t^{12}+1800z^{4}t^{14}-6687z^{2}t^{16}+5039360zw^{17}-49127872zw^{13}t^{4}+66343424zw^{11}t^{6}-23619888zw^{9}t^{8}-4654992zw^{7}t^{10}+4117020zw^{5}t^{12}-559464zw^{3}t^{14}-1634zwt^{16}-256w^{18}-2519680w^{16}t^{2}-11338176w^{14}t^{4}+51648320w^{12}t^{6}-50493856w^{10}t^{8}+9659808w^{8}t^{10}+8402772w^{6}t^{12}-4204620w^{4}t^{14}+497733w^{2}t^{16}+11328t^{18}}{t^{12}(116yw^{5}-158yw^{3}t^{2}+50ywt^{4}-4z^{6}-6z^{2}t^{4}+116zw^{5}-11zwt^{4}-4w^{6}-58w^{4}t^{2}-27w^{2}t^{4}+23t^{6})}$

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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
12.72.3.ce.1 $12$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.1.ig.1 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.72.1.n.1 $24$ $2$ $2$ $1$ $1$ $1^{4}$
24.72.1.bn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.cf.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.3.ls.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.qe.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.tf.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
72.432.21.lx.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.og.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.ry.1 $72$ $3$ $3$ $21$ $?$ not computed