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Properties

Label 24.144.5.ih.1
Level $24$
Index $144$
Genus $5$
Analytic rank $0$
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: $0$
$\Q$-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: 24.144.5.577

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&0\\0&17\end{bmatrix}$, $\begin{bmatrix}17&6\\12&13\end{bmatrix}$, $\begin{bmatrix}19&3\\18&11\end{bmatrix}$, $\begin{bmatrix}23&0\\12&19\end{bmatrix}$, $\begin{bmatrix}23&3\\6&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.ih.1.1, 24.288.5-24.ih.1.2, 24.288.5-24.ih.1.3, 24.288.5-24.ih.1.4, 24.288.5-24.ih.1.5, 24.288.5-24.ih.1.6, 120.288.5-24.ih.1.1, 120.288.5-24.ih.1.2, 120.288.5-24.ih.1.3, 120.288.5-24.ih.1.4, 120.288.5-24.ih.1.5, 120.288.5-24.ih.1.6, 168.288.5-24.ih.1.1, 168.288.5-24.ih.1.2, 168.288.5-24.ih.1.3, 168.288.5-24.ih.1.4, 168.288.5-24.ih.1.5, 168.288.5-24.ih.1.6, 264.288.5-24.ih.1.1, 264.288.5-24.ih.1.2, 264.288.5-24.ih.1.3, 264.288.5-24.ih.1.4, 264.288.5-24.ih.1.5, 264.288.5-24.ih.1.6, 312.288.5-24.ih.1.1, 312.288.5-24.ih.1.2, 312.288.5-24.ih.1.3, 312.288.5-24.ih.1.4, 312.288.5-24.ih.1.5, 312.288.5-24.ih.1.6
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{21}\cdot3^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 24.2.a.a$^{2}$, 72.2.a.a, 576.2.a.f$^{2}$

Models

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

$ 0 $ $=$ $ x^{2} - y^{2} + y z - z^{2} + t^{2} $
$=$ $x^{2} + 2 x y + 2 x z + y^{2} - y z + z^{2}$
$=$ $4 x^{2} - 2 x y - 2 x z - 6 y z - w^{2} + t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4320 x^{8} - 864 x^{7} y - 396 x^{6} y^{2} + 2304 x^{6} z^{2} + 36 x^{5} y^{3} - 720 x^{5} y z^{2} + \cdots + 43 z^{8} $
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Rational points

This modular curve has no $\Q_p$ points for $p=7,31,37$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{(w^{6}+16t^{6})^{3}}{t^{12}w^{6}}$

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
12.72.3.cy.1 $12$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.1.ja.1 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.72.1.y.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.bn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.cn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.3.tg.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.tx.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.uy.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
24.288.17.efe.1 $24$ $2$ $2$ $17$ $3$ $1^{12}$
24.288.17.efm.1 $24$ $2$ $2$ $17$ $2$ $1^{12}$
24.288.17.gix.1 $24$ $2$ $2$ $17$ $4$ $1^{12}$
24.288.17.gjq.1 $24$ $2$ $2$ $17$ $3$ $1^{12}$
72.432.21.ut.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.vv.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.yi.1 $72$ $3$ $3$ $21$ $?$ not computed
120.288.17.bvsg.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.bvsk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cifc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.cifg.1 $120$ $2$ $2$ $17$ $?$ not computed
168.288.17.bovu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.bovy.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.bxeg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.bxek.1 $168$ $2$ $2$ $17$ $?$ not computed
264.288.17.bozd.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.bozh.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.bxhp.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.bxht.1 $264$ $2$ $2$ $17$ $?$ not computed
312.288.17.bozg.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.bozk.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.bxhq.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.bxhu.1 $312$ $2$ $2$ $17$ $?$ not computed