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Properties

Label 24.96.1.cn.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1330

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&10\\12&17\end{bmatrix}$, $\begin{bmatrix}1&14\\12&13\end{bmatrix}$, $\begin{bmatrix}7&2\\18&23\end{bmatrix}$, $\begin{bmatrix}11&8\\12&11\end{bmatrix}$, $\begin{bmatrix}13&16\\6&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cn.2.1, 24.192.1-24.cn.2.2, 24.192.1-24.cn.2.3, 24.192.1-24.cn.2.4, 24.192.1-24.cn.2.5, 24.192.1-24.cn.2.6, 24.192.1-24.cn.2.7, 24.192.1-24.cn.2.8, 24.192.1-24.cn.2.9, 24.192.1-24.cn.2.10, 24.192.1-24.cn.2.11, 24.192.1-24.cn.2.12, 24.192.1-24.cn.2.13, 24.192.1-24.cn.2.14, 24.192.1-24.cn.2.15, 24.192.1-24.cn.2.16, 120.192.1-24.cn.2.1, 120.192.1-24.cn.2.2, 120.192.1-24.cn.2.3, 120.192.1-24.cn.2.4, 120.192.1-24.cn.2.5, 120.192.1-24.cn.2.6, 120.192.1-24.cn.2.7, 120.192.1-24.cn.2.8, 120.192.1-24.cn.2.9, 120.192.1-24.cn.2.10, 120.192.1-24.cn.2.11, 120.192.1-24.cn.2.12, 120.192.1-24.cn.2.13, 120.192.1-24.cn.2.14, 120.192.1-24.cn.2.15, 120.192.1-24.cn.2.16, 168.192.1-24.cn.2.1, 168.192.1-24.cn.2.2, 168.192.1-24.cn.2.3, 168.192.1-24.cn.2.4, 168.192.1-24.cn.2.5, 168.192.1-24.cn.2.6, 168.192.1-24.cn.2.7, 168.192.1-24.cn.2.8, 168.192.1-24.cn.2.9, 168.192.1-24.cn.2.10, 168.192.1-24.cn.2.11, 168.192.1-24.cn.2.12, 168.192.1-24.cn.2.13, 168.192.1-24.cn.2.14, 168.192.1-24.cn.2.15, 168.192.1-24.cn.2.16, 264.192.1-24.cn.2.1, 264.192.1-24.cn.2.2, 264.192.1-24.cn.2.3, 264.192.1-24.cn.2.4, 264.192.1-24.cn.2.5, 264.192.1-24.cn.2.6, 264.192.1-24.cn.2.7, 264.192.1-24.cn.2.8, 264.192.1-24.cn.2.9, 264.192.1-24.cn.2.10, 264.192.1-24.cn.2.11, 264.192.1-24.cn.2.12, 264.192.1-24.cn.2.13, 264.192.1-24.cn.2.14, 264.192.1-24.cn.2.15, 264.192.1-24.cn.2.16, 312.192.1-24.cn.2.1, 312.192.1-24.cn.2.2, 312.192.1-24.cn.2.3, 312.192.1-24.cn.2.4, 312.192.1-24.cn.2.5, 312.192.1-24.cn.2.6, 312.192.1-24.cn.2.7, 312.192.1-24.cn.2.8, 312.192.1-24.cn.2.9, 312.192.1-24.cn.2.10, 312.192.1-24.cn.2.11, 312.192.1-24.cn.2.12, 312.192.1-24.cn.2.13, 312.192.1-24.cn.2.14, 312.192.1-24.cn.2.15, 312.192.1-24.cn.2.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 24x - 56 $
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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.

Weierstrass model
$(0:1:0)$, $(2:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{48x^{2}y^{30}-93312x^{2}y^{28}z^{2}-33592320x^{2}y^{26}z^{4}+3851937275904x^{2}y^{24}z^{6}+1266051435134976x^{2}y^{22}z^{8}+12862342823262289920x^{2}y^{20}z^{10}-24244030489958874611712x^{2}y^{18}z^{12}-12008584070097916775104512x^{2}y^{16}z^{14}-1143666561233807533878018048x^{2}y^{14}z^{16}+68949369250643486876689760256x^{2}y^{12}z^{18}+5983416926391023541696470188032x^{2}y^{10}z^{20}+59046643829758819156652351029248x^{2}y^{8}z^{22}+15212703877594496914906822974898176x^{2}y^{6}z^{24}+129290407179696595904137318285443072x^{2}y^{4}z^{26}-29405065906854270953324094403897196544x^{2}y^{2}z^{28}-454034544748725490855096936058918535168x^{2}z^{30}+96xy^{30}z+5412096xy^{28}z^{3}-12241041408xy^{26}z^{5}+27939243147264xy^{24}z^{7}-93709121252622336xy^{22}z^{9}-227442078550645014528xy^{20}z^{11}+158119931506129958338560xy^{18}z^{13}+53359158646669320589934592xy^{16}z^{15}+2326626964717237267184222208xy^{14}z^{17}+247171155540851057896080801792xy^{12}z^{19}+38872583963244525196446145708032xy^{10}z^{21}-1321470352616209426794821883789312xy^{8}z^{23}-159535601304806363096888247941332992xy^{6}z^{25}-499837355796444904264630128756129792xy^{4}z^{27}+29612957181922368705547124136524906496xy^{2}z^{29}-908069089497450981710193872117837070336xz^{31}+y^{32}-21120y^{30}z^{2}+37698048y^{28}z^{4}-202575126528y^{26}z^{6}-334319905751040y^{24}z^{8}+825236085435531264y^{22}z^{10}+2104740368088985239552y^{20}z^{12}+161890600828535706746880y^{18}z^{14}+18771979010690013154246656y^{16}z^{16}-10800148763885844592739745792y^{14}z^{18}-2545849726065150540621501431808y^{12}z^{20}-93512103286048304970564693393408y^{10}z^{22}+4954788374433291516147167970459648y^{8}z^{24}+270340248814453057422323091001835520y^{6}z^{26}+3042655636953455373585206765804322816y^{4}z^{28}+136561468689177101238061308810557915136y^{2}z^{30}+3654728615697158484080862699595140956160z^{32}}{z^{4}y^{4}(y^{2}+216z^{2})^{6}(24x^{2}y^{10}+104976x^{2}y^{8}z^{2}-110854656x^{2}y^{6}z^{4}-48977602560x^{2}y^{4}z^{6}+228509902503936x^{2}z^{10}-456xy^{10}z+1539648xy^{8}z^{3}+1169012736xy^{6}z^{5}-104485552128xy^{4}z^{7}-38084983750656xy^{2}z^{9}-914039610015744xz^{11}+y^{12}-9552y^{10}z^{2}-13110336y^{8}z^{4}+3184551936y^{6}z^{6}+1286478360576y^{4}z^{8}+76169967501312y^{2}z^{10}+914039610015744z^{12})}$

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.a.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.by.1 $24$ $2$ $2$ $1$ $0$ dimension zero

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.bl.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bm.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bn.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bo.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bv.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bx.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.by.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ca.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.g.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.f.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.n.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.z.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.jx.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jy.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ka.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ke.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kh.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ki.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.jx.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jy.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ka.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kb.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ke.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kh.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ki.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.jx.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jy.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ka.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kb.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ke.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kf.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kh.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ki.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.jx.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jy.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ka.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kb.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ke.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kf.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kh.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ki.1 $312$ $2$ $2$ $5$ $?$ not computed