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Properties

Label 24.96.3.gs.1
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: $144$
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: 24.96.3.122

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\8&19\end{bmatrix}$, $\begin{bmatrix}11&12\\4&1\end{bmatrix}$, $\begin{bmatrix}23&3\\8&13\end{bmatrix}$, $\begin{bmatrix}23&9\\12&17\end{bmatrix}$, $\begin{bmatrix}23&18\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035917
Contains $-I$: yes
Quadratic refinements: 24.192.3-24.gs.1.1, 24.192.3-24.gs.1.2, 24.192.3-24.gs.1.3, 24.192.3-24.gs.1.4, 24.192.3-24.gs.1.5, 24.192.3-24.gs.1.6, 24.192.3-24.gs.1.7, 24.192.3-24.gs.1.8, 24.192.3-24.gs.1.9, 24.192.3-24.gs.1.10, 24.192.3-24.gs.1.11, 24.192.3-24.gs.1.12, 24.192.3-24.gs.1.13, 24.192.3-24.gs.1.14, 24.192.3-24.gs.1.15, 24.192.3-24.gs.1.16, 120.192.3-24.gs.1.1, 120.192.3-24.gs.1.2, 120.192.3-24.gs.1.3, 120.192.3-24.gs.1.4, 120.192.3-24.gs.1.5, 120.192.3-24.gs.1.6, 120.192.3-24.gs.1.7, 120.192.3-24.gs.1.8, 120.192.3-24.gs.1.9, 120.192.3-24.gs.1.10, 120.192.3-24.gs.1.11, 120.192.3-24.gs.1.12, 120.192.3-24.gs.1.13, 120.192.3-24.gs.1.14, 120.192.3-24.gs.1.15, 120.192.3-24.gs.1.16, 168.192.3-24.gs.1.1, 168.192.3-24.gs.1.2, 168.192.3-24.gs.1.3, 168.192.3-24.gs.1.4, 168.192.3-24.gs.1.5, 168.192.3-24.gs.1.6, 168.192.3-24.gs.1.7, 168.192.3-24.gs.1.8, 168.192.3-24.gs.1.9, 168.192.3-24.gs.1.10, 168.192.3-24.gs.1.11, 168.192.3-24.gs.1.12, 168.192.3-24.gs.1.13, 168.192.3-24.gs.1.14, 168.192.3-24.gs.1.15, 168.192.3-24.gs.1.16, 264.192.3-24.gs.1.1, 264.192.3-24.gs.1.2, 264.192.3-24.gs.1.3, 264.192.3-24.gs.1.4, 264.192.3-24.gs.1.5, 264.192.3-24.gs.1.6, 264.192.3-24.gs.1.7, 264.192.3-24.gs.1.8, 264.192.3-24.gs.1.9, 264.192.3-24.gs.1.10, 264.192.3-24.gs.1.11, 264.192.3-24.gs.1.12, 264.192.3-24.gs.1.13, 264.192.3-24.gs.1.14, 264.192.3-24.gs.1.15, 264.192.3-24.gs.1.16, 312.192.3-24.gs.1.1, 312.192.3-24.gs.1.2, 312.192.3-24.gs.1.3, 312.192.3-24.gs.1.4, 312.192.3-24.gs.1.5, 312.192.3-24.gs.1.6, 312.192.3-24.gs.1.7, 312.192.3-24.gs.1.8, 312.192.3-24.gs.1.9, 312.192.3-24.gs.1.10, 312.192.3-24.gs.1.11, 312.192.3-24.gs.1.12, 312.192.3-24.gs.1.13, 312.192.3-24.gs.1.14, 312.192.3-24.gs.1.15, 312.192.3-24.gs.1.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{12}\cdot3^{4}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 48.2.c.a, 144.2.a.b

Models

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

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

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

$ y^{2} $ $=$ $ 3x^{7} + 15x^{6} + 21x^{5} + 30x^{4} + 21x^{3} + 15x^{2} + 3x $
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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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}t$
$\displaystyle Z$ $=$ $\displaystyle 2z$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Y$ $=$ $\displaystyle -\frac{1}{8}y^{3}t-\frac{1}{2}y^{2}zt-\frac{1}{4}yz^{2}t+\frac{1}{2}z^{3}t$
$\displaystyle Z$ $=$ $\displaystyle -\frac{1}{2}y-z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3}\cdot\frac{2583252xzw^{12}-10582056xzw^{10}t^{2}-12157776xzw^{8}t^{4}+46185984xzw^{6}t^{6}+22554216xzw^{4}t^{8}+1220292xzw^{2}t^{10}-21888xzt^{12}-4536xw^{13}-267300xw^{11}t^{2}+102816xw^{9}t^{4}+3401568xw^{7}t^{6}-2245464xw^{5}t^{8}-2448996xw^{3}t^{10}-242880xwt^{12}+1409940yzw^{12}-6014952yzw^{10}t^{2}-6692544yzw^{8}t^{4}+31065696yzw^{6}t^{6}+20099976yzw^{4}t^{8}+2376720yzw^{2}t^{10}+25920yzt^{12}-704970yw^{13}+4345380yw^{11}t^{2}-807552yw^{9}t^{4}-17690400yw^{7}t^{6}-3368448yw^{5}t^{8}+2246148yw^{3}t^{10}+254496ywt^{12}-467748z^{2}w^{12}+3039552z^{2}w^{10}t^{2}-1499472z^{2}w^{8}t^{4}-10284336z^{2}w^{6}t^{6}-188736z^{2}w^{4}t^{8}+994104z^{2}w^{2}t^{10}+37952z^{2}t^{12}-233280zw^{13}+1664064zw^{11}t^{2}+48960zw^{9}t^{4}-10004976zw^{7}t^{6}-2513208zw^{5}t^{8}+1789836zw^{3}t^{10}+283072zwt^{12}-2619w^{14}-175392w^{12}t^{2}+465012w^{10}t^{4}+1378836w^{8}t^{6}-3024432w^{6}t^{8}-2175468w^{4}t^{10}-196756w^{2}t^{12}}{t^{2}(324xzw^{10}-3348xzw^{8}t^{2}+8460xzw^{6}t^{4}+120xzw^{4}t^{6}+12xzt^{10}+648xw^{7}t^{4}-252xw^{5}t^{6}+564xw^{3}t^{8}+48xwt^{10}+324yzw^{10}-2376yzw^{8}t^{2}+4320yzw^{6}t^{4}+1092yzw^{4}t^{6}-504yzw^{2}t^{8}-162yw^{11}+1188yw^{9}t^{2}-2520yw^{7}t^{4}+474yw^{5}t^{6}-44yw^{3}t^{8}-60ywt^{10}-108z^{2}w^{8}t^{2}-504z^{2}w^{6}t^{4}-564z^{2}w^{4}t^{6}-432z^{2}w^{2}t^{8}+24z^{2}t^{10}-108zw^{9}t^{2}+36zw^{7}t^{4}+780zw^{5}t^{6}-1016zw^{3}t^{8}+60zwt^{10}-27w^{10}t^{2}+504w^{8}t^{4}-273w^{6}t^{6}+248w^{4}t^{8}+60w^{2}t^{10})}$

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
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iu.1 $24$ $2$ $2$ $1$ $0$ $2$
24.48.2.f.1 $24$ $2$ $2$ $2$ $0$ $1$

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.cx.2 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.di.2 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.192.5.eh.2 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.el.2 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.ev.2 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.ex.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.192.5.fr.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.fw.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.288.13.lb.2 $24$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
72.288.13.fk.3 $72$ $3$ $3$ $13$ $?$ not computed
72.288.19.ko.1 $72$ $3$ $3$ $19$ $?$ not computed
72.288.19.lk.1 $72$ $3$ $3$ $19$ $?$ not computed
120.192.5.bhb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bhd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bhr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bht.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bjn.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bjp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bkd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bkf.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bhb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bhd.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bhr.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bht.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bjn.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bjp.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bkd.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bkf.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bhb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bhd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bhr.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bht.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bjn.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bjp.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bkd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bkf.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bhb.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bhd.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bhr.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bht.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bjn.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bjp.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bkd.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bkf.3 $312$ $2$ $2$ $5$ $?$ not computed