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Properties

Label 24.96.1.dh.4
Level $24$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ 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 $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1604

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&6\\12&23\end{bmatrix}$, $\begin{bmatrix}13&3\\4&11\end{bmatrix}$, $\begin{bmatrix}13&9\\0&11\end{bmatrix}$, $\begin{bmatrix}17&18\\12&1\end{bmatrix}$, $\begin{bmatrix}17&18\\12&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035916
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dh.4.1, 24.192.1-24.dh.4.2, 24.192.1-24.dh.4.3, 24.192.1-24.dh.4.4, 24.192.1-24.dh.4.5, 24.192.1-24.dh.4.6, 24.192.1-24.dh.4.7, 24.192.1-24.dh.4.8, 24.192.1-24.dh.4.9, 24.192.1-24.dh.4.10, 24.192.1-24.dh.4.11, 24.192.1-24.dh.4.12, 24.192.1-24.dh.4.13, 24.192.1-24.dh.4.14, 24.192.1-24.dh.4.15, 24.192.1-24.dh.4.16, 120.192.1-24.dh.4.1, 120.192.1-24.dh.4.2, 120.192.1-24.dh.4.3, 120.192.1-24.dh.4.4, 120.192.1-24.dh.4.5, 120.192.1-24.dh.4.6, 120.192.1-24.dh.4.7, 120.192.1-24.dh.4.8, 120.192.1-24.dh.4.9, 120.192.1-24.dh.4.10, 120.192.1-24.dh.4.11, 120.192.1-24.dh.4.12, 120.192.1-24.dh.4.13, 120.192.1-24.dh.4.14, 120.192.1-24.dh.4.15, 120.192.1-24.dh.4.16, 168.192.1-24.dh.4.1, 168.192.1-24.dh.4.2, 168.192.1-24.dh.4.3, 168.192.1-24.dh.4.4, 168.192.1-24.dh.4.5, 168.192.1-24.dh.4.6, 168.192.1-24.dh.4.7, 168.192.1-24.dh.4.8, 168.192.1-24.dh.4.9, 168.192.1-24.dh.4.10, 168.192.1-24.dh.4.11, 168.192.1-24.dh.4.12, 168.192.1-24.dh.4.13, 168.192.1-24.dh.4.14, 168.192.1-24.dh.4.15, 168.192.1-24.dh.4.16, 264.192.1-24.dh.4.1, 264.192.1-24.dh.4.2, 264.192.1-24.dh.4.3, 264.192.1-24.dh.4.4, 264.192.1-24.dh.4.5, 264.192.1-24.dh.4.6, 264.192.1-24.dh.4.7, 264.192.1-24.dh.4.8, 264.192.1-24.dh.4.9, 264.192.1-24.dh.4.10, 264.192.1-24.dh.4.11, 264.192.1-24.dh.4.12, 264.192.1-24.dh.4.13, 264.192.1-24.dh.4.14, 264.192.1-24.dh.4.15, 264.192.1-24.dh.4.16, 312.192.1-24.dh.4.1, 312.192.1-24.dh.4.2, 312.192.1-24.dh.4.3, 312.192.1-24.dh.4.4, 312.192.1-24.dh.4.5, 312.192.1-24.dh.4.6, 312.192.1-24.dh.4.7, 312.192.1-24.dh.4.8, 312.192.1-24.dh.4.9, 312.192.1-24.dh.4.10, 312.192.1-24.dh.4.11, 312.192.1-24.dh.4.12, 312.192.1-24.dh.4.13, 312.192.1-24.dh.4.14, 312.192.1-24.dh.4.15, 312.192.1-24.dh.4.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 24x + 56 $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

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^{12}\cdot3^{12}}\cdot\frac{48x^{2}y^{30}+93312x^{2}y^{28}z^{2}-839808000x^{2}y^{26}z^{4}+719305629696x^{2}y^{24}z^{6}-144503518593024x^{2}y^{22}z^{8}-128755456176291840x^{2}y^{20}z^{10}+114759501116696690688x^{2}y^{18}z^{12}-48701083395269578457088x^{2}y^{16}z^{14}+12713797364680323017736192x^{2}y^{14}z^{16}-1946285661683620764425650176x^{2}y^{12}z^{18}+104194791251418280378459226112x^{2}y^{10}z^{20}+24500197038014518767340436324352x^{2}y^{8}z^{22}-6442637275332352274992107507154944x^{2}y^{6}z^{24}+650301874325669974932224179068469248x^{2}y^{4}z^{26}-29405065906854270953324094403897196544x^{2}y^{2}z^{28}+454034544748725490855096936058918535168x^{2}z^{30}-96xy^{30}z+5412096xy^{28}z^{3}-3077056512xy^{26}z^{5}-3928850251776xy^{24}z^{7}+5126270158503936xy^{22}z^{9}-2962075127311761408xy^{20}z^{11}+924715592660727889920xy^{18}z^{13}-137545733488799895257088xy^{16}z^{15}-15503832110785504191971328xy^{14}z^{17}+14318417377397026687404736512xy^{12}z^{19}-3906683896361609694923238408192xy^{10}z^{21}+616816355516132013041810782814208xy^{8}z^{23}-57017810224462128802101056879198208xy^{6}z^{25}+2618531770225021379080815860659519488xy^{4}z^{27}-29612957181922368705547124136524906496xy^{2}z^{29}-908069089497450981710193872117837070336xz^{31}-y^{32}-21120y^{30}z^{2}-4105728y^{28}z^{4}+74763067392y^{26}z^{6}-72128892026880y^{24}z^{8}+33519801007079424y^{22}z^{10}-7771006782199037952y^{20}z^{12}-885082835170445230080y^{18}z^{14}+1265687992804440375558144y^{16}z^{16}-480745708240116236138053632y^{14}z^{18}+110410996853663576654409105408y^{12}z^{20}-16772782785278720507045392416768y^{10}z^{22}+1595283949600138177093866558062592y^{8}z^{24}-76145209632376529616059796711014400y^{6}z^{26}+75713489068010909760239223611326464y^{4}z^{28}+136561468689177101238061308810557915136y^{2}z^{30}-3654728615697158484080862699595140956160z^{32}}{z^{8}y^{8}(y^{2}-216z^{2})^{3}(38880x^{2}y^{6}z^{2}-22954752x^{2}y^{4}z^{4}+2176782336x^{2}y^{2}z^{6}-432xy^{8}z+435456xy^{6}z^{3}-29673216xy^{4}z^{5}-10883911680xy^{2}z^{7}+705277476864xz^{9}+y^{10}-2808y^{8}z^{2}-590976y^{6}z^{4}+485968896y^{4}z^{6}-50065993728y^{2}z^{8}+1410554953728z^{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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bs.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.is.1 $24$ $2$ $2$ $1$ $1$ 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.cw.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.di.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dw.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.dz.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fc.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fg.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fm.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fn.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.288.9.z.2 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.v.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.es.4 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fi.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zr.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zt.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bab.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcf.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcn.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zr.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zt.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zz.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bab.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcd.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcl.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcn.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.zr.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zt.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zz.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bab.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcd.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcf.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcl.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcn.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zr.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zt.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zz.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bab.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcd.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcf.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcl.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcn.2 $312$ $2$ $2$ $5$ $?$ not computed