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Properties

Label 24.144.5.hd.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: 24.144.5.588

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&16\\18&23\end{bmatrix}$, $\begin{bmatrix}1&21\\6&19\end{bmatrix}$, $\begin{bmatrix}11&8\\6&13\end{bmatrix}$, $\begin{bmatrix}17&9\\18&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.hd.1.1, 24.288.5-24.hd.1.2, 24.288.5-24.hd.1.3, 24.288.5-24.hd.1.4, 120.288.5-24.hd.1.1, 120.288.5-24.hd.1.2, 120.288.5-24.hd.1.3, 120.288.5-24.hd.1.4, 168.288.5-24.hd.1.1, 168.288.5-24.hd.1.2, 168.288.5-24.hd.1.3, 168.288.5-24.hd.1.4, 264.288.5-24.hd.1.1, 264.288.5-24.hd.1.2, 264.288.5-24.hd.1.3, 264.288.5-24.hd.1.4, 312.288.5-24.hd.1.1, 312.288.5-24.hd.1.2, 312.288.5-24.hd.1.3, 312.288.5-24.hd.1.4
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{21}\cdot3^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 24.2.a.a, 72.2.a.a$^{2}$, 576.2.a.f$^{2}$

Models

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

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

$ 0 $ $=$ $ - 27 x^{8} - 54 x^{7} y - 63 x^{6} y^{2} + 36 x^{6} z^{2} - 18 x^{5} y^{3} + 24 x^{5} y z^{2} + \cdots + 32 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{27715712xw^{15}t^{2}+66772352xw^{13}t^{4}+22674048xw^{11}t^{6}-41684912xw^{9}t^{8}-37193568xw^{7}t^{10}-9947796xw^{5}t^{12}-355122xw^{3}t^{14}+133400xwt^{16}-256z^{18}-2304z^{16}t^{2}-8640z^{14}t^{4}-18240z^{12}t^{6}-25488z^{10}t^{8}-25920z^{8}t^{10}-22248z^{6}t^{12}-19908z^{4}t^{14}-11178z^{2}t^{16}-5039360zw^{17}+49134592zw^{13}t^{4}+66353600zw^{11}t^{6}+23618160zw^{9}t^{8}-4651248zw^{7}t^{10}-4113312zw^{5}t^{12}-560640zw^{3}t^{14}+688zwt^{16}-256w^{18}+2519680w^{16}t^{2}-11339328w^{14}t^{4}-51655232w^{12}t^{6}-50499952w^{10}t^{8}-9658224w^{8}t^{10}+8398728w^{6}t^{12}+4202682w^{4}t^{14}+492249w^{2}t^{16}-11523t^{18}}{t^{12}(170xw^{3}t^{2}+68xwt^{4}-4z^{6}-12z^{4}t^{2}-9z^{2}t^{4}-116zw^{5}+10zwt^{4}-4w^{6}+58w^{4}t^{2}-33w^{2}t^{4}-23t^{6})}$

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.cf.1 $12$ $2$ $2$ $3$ $0$ $1^{2}$
24.48.1.im.1 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.72.1.t.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.bn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.cc.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.3.mu.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.qk.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.tg.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.mj.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.pi.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.tg.1 $72$ $3$ $3$ $21$ $?$ not computed