Properties

Label 24.96.3.gm.2
Level $24$
Index $96$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24W3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.3.118

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&0\\16&19\end{bmatrix}$, $\begin{bmatrix}23&3\\8&11\end{bmatrix}$, $\begin{bmatrix}23&6\\0&7\end{bmatrix}$, $\begin{bmatrix}23&12\\8&17\end{bmatrix}$, $\begin{bmatrix}23&15\\12&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.3-24.gm.2.1, 24.192.3-24.gm.2.2, 24.192.3-24.gm.2.3, 24.192.3-24.gm.2.4, 24.192.3-24.gm.2.5, 24.192.3-24.gm.2.6, 24.192.3-24.gm.2.7, 24.192.3-24.gm.2.8, 24.192.3-24.gm.2.9, 24.192.3-24.gm.2.10, 24.192.3-24.gm.2.11, 24.192.3-24.gm.2.12, 24.192.3-24.gm.2.13, 24.192.3-24.gm.2.14, 24.192.3-24.gm.2.15, 24.192.3-24.gm.2.16, 120.192.3-24.gm.2.1, 120.192.3-24.gm.2.2, 120.192.3-24.gm.2.3, 120.192.3-24.gm.2.4, 120.192.3-24.gm.2.5, 120.192.3-24.gm.2.6, 120.192.3-24.gm.2.7, 120.192.3-24.gm.2.8, 120.192.3-24.gm.2.9, 120.192.3-24.gm.2.10, 120.192.3-24.gm.2.11, 120.192.3-24.gm.2.12, 120.192.3-24.gm.2.13, 120.192.3-24.gm.2.14, 120.192.3-24.gm.2.15, 120.192.3-24.gm.2.16, 168.192.3-24.gm.2.1, 168.192.3-24.gm.2.2, 168.192.3-24.gm.2.3, 168.192.3-24.gm.2.4, 168.192.3-24.gm.2.5, 168.192.3-24.gm.2.6, 168.192.3-24.gm.2.7, 168.192.3-24.gm.2.8, 168.192.3-24.gm.2.9, 168.192.3-24.gm.2.10, 168.192.3-24.gm.2.11, 168.192.3-24.gm.2.12, 168.192.3-24.gm.2.13, 168.192.3-24.gm.2.14, 168.192.3-24.gm.2.15, 168.192.3-24.gm.2.16, 264.192.3-24.gm.2.1, 264.192.3-24.gm.2.2, 264.192.3-24.gm.2.3, 264.192.3-24.gm.2.4, 264.192.3-24.gm.2.5, 264.192.3-24.gm.2.6, 264.192.3-24.gm.2.7, 264.192.3-24.gm.2.8, 264.192.3-24.gm.2.9, 264.192.3-24.gm.2.10, 264.192.3-24.gm.2.11, 264.192.3-24.gm.2.12, 264.192.3-24.gm.2.13, 264.192.3-24.gm.2.14, 264.192.3-24.gm.2.15, 264.192.3-24.gm.2.16, 312.192.3-24.gm.2.1, 312.192.3-24.gm.2.2, 312.192.3-24.gm.2.3, 312.192.3-24.gm.2.4, 312.192.3-24.gm.2.5, 312.192.3-24.gm.2.6, 312.192.3-24.gm.2.7, 312.192.3-24.gm.2.8, 312.192.3-24.gm.2.9, 312.192.3-24.gm.2.10, 312.192.3-24.gm.2.11, 312.192.3-24.gm.2.12, 312.192.3-24.gm.2.13, 312.192.3-24.gm.2.14, 312.192.3-24.gm.2.15, 312.192.3-24.gm.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{18}\cdot3^{3}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 192.2.a.b, 192.2.c.a

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

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

$ 0 $ $=$ $ x^{7} + 6 x^{6} z + 2 x^{5} y^{2} + 16 x^{5} z^{2} + 12 x^{4} y^{2} z + 24 x^{4} z^{3} + \cdots + 4 y^{2} z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -2x^{7} - 10x^{6} - 14x^{5} - 20x^{4} - 14x^{3} - 10x^{2} - 2x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:0:1)$, $(0:0:1/2:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 4y^{3}t+8y^{2}wt+5yw^{2}t+w^{3}t$
$\displaystyle Z$ $=$ $\displaystyle -y-w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{43322x^{2}w^{12}-56884x^{2}w^{10}t^{2}+4316x^{2}w^{8}t^{4}+44520x^{2}w^{6}t^{6}-5586x^{2}w^{4}t^{8}+684x^{2}w^{2}t^{10}-158x^{2}t^{12}-41850xzw^{12}+54516xzw^{10}t^{2}+77920xzw^{8}t^{4}-62056xzw^{6}t^{6}+7026xzw^{4}t^{8}-2628xzw^{2}t^{10}+1134xzt^{12}-19901xw^{13}-79625xw^{11}t^{2}-31836xw^{9}t^{4}-21776xw^{7}t^{6}+21081xw^{5}t^{8}-6369xw^{3}t^{10}+1697xwt^{12}+20736yw^{13}+10686yw^{11}t^{2}-13860yw^{9}t^{4}-20404yw^{7}t^{6}+2677yw^{5}t^{8}-5514yw^{3}t^{10}+510ywt^{12}+2428z^{2}w^{12}-122448z^{2}w^{10}t^{2}-108224z^{2}w^{8}t^{4}+10240z^{2}w^{6}t^{6}+9764z^{2}w^{4}t^{8}+1248z^{2}w^{2}t^{10}-972z^{2}t^{12}-256zw^{13}-37700zw^{11}t^{2}-14876zw^{9}t^{4}+33152zw^{7}t^{6}-16984zw^{5}t^{8}+16860zw^{3}t^{10}-2920zwt^{12}-735w^{14}+85998w^{12}t^{2}-31078w^{10}t^{4}-3202w^{8}t^{6}-12656w^{6}t^{8}+9870w^{4}t^{10}-1019w^{2}t^{12}+2t^{14}}{t^{2}w^{3}(4x^{2}w^{7}+2x^{2}w^{5}t^{2}-120x^{2}w^{3}t^{4}-122x^{2}wt^{6}-4xzw^{7}+42xzw^{5}t^{2}+120xzw^{3}t^{4}+122xzwt^{6}-41xw^{6}t^{2}-48xw^{4}t^{4}+477xw^{2}t^{6}-81xt^{8}-2yw^{6}t^{2}+18yw^{4}t^{4}+57yw^{2}t^{6}-122yt^{8}-84z^{2}w^{5}t^{2}+244z^{2}wt^{6}+4zw^{6}t^{2}-344zw^{2}t^{6}+324zt^{8}+21w^{7}t^{2}-20w^{5}t^{4}-176w^{3}t^{6}+202wt^{8})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iq.1 $24$ $2$ $2$ $1$ $0$ $2$
24.48.2.g.2 $24$ $2$ $2$ $2$ $0$ $1$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.192.5.de.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.dl.4 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.dq.2 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.du.4 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.eu.4 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.ez.4 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.192.5.fc.4 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.192.5.fh.4 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.288.13.ku.1 $24$ $3$ $3$ $13$ $1$ $1^{4}\cdot2^{3}$
72.288.13.fa.4 $72$ $3$ $3$ $13$ $?$ not computed
72.288.19.ki.3 $72$ $3$ $3$ $19$ $?$ not computed
72.288.19.le.3 $72$ $3$ $3$ $19$ $?$ not computed
120.192.5.bfz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bgb.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bgh.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bgj.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bil.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bin.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bit.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.biv.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bfz.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bgb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bgh.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bgj.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bil.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bin.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bit.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.biv.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bfz.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bgb.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bgh.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bgj.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bil.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bin.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bit.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.biv.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bfz.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bgb.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bgh.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bgj.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bil.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bin.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bit.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.biv.3 $312$ $2$ $2$ $5$ $?$ not computed