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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ 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 $4^{8}\cdot8^{8}$ Cusp orbits $1^{2}\cdot2^{3}\cdot8$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.307

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&4\\0&5\end{bmatrix}$, $\begin{bmatrix}15&20\\16&15\end{bmatrix}$, $\begin{bmatrix}17&8\\22&11\end{bmatrix}$, $\begin{bmatrix}19&12\\14&1\end{bmatrix}$, $\begin{bmatrix}19&12\\20&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^4\times \GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.n.2.1, 24.192.1-24.n.2.2, 24.192.1-24.n.2.3, 24.192.1-24.n.2.4, 24.192.1-24.n.2.5, 24.192.1-24.n.2.6, 24.192.1-24.n.2.7, 24.192.1-24.n.2.8, 24.192.1-24.n.2.9, 24.192.1-24.n.2.10, 24.192.1-24.n.2.11, 24.192.1-24.n.2.12, 24.192.1-24.n.2.13, 24.192.1-24.n.2.14, 24.192.1-24.n.2.15, 24.192.1-24.n.2.16, 120.192.1-24.n.2.1, 120.192.1-24.n.2.2, 120.192.1-24.n.2.3, 120.192.1-24.n.2.4, 120.192.1-24.n.2.5, 120.192.1-24.n.2.6, 120.192.1-24.n.2.7, 120.192.1-24.n.2.8, 120.192.1-24.n.2.9, 120.192.1-24.n.2.10, 120.192.1-24.n.2.11, 120.192.1-24.n.2.12, 120.192.1-24.n.2.13, 120.192.1-24.n.2.14, 120.192.1-24.n.2.15, 120.192.1-24.n.2.16, 168.192.1-24.n.2.1, 168.192.1-24.n.2.2, 168.192.1-24.n.2.3, 168.192.1-24.n.2.4, 168.192.1-24.n.2.5, 168.192.1-24.n.2.6, 168.192.1-24.n.2.7, 168.192.1-24.n.2.8, 168.192.1-24.n.2.9, 168.192.1-24.n.2.10, 168.192.1-24.n.2.11, 168.192.1-24.n.2.12, 168.192.1-24.n.2.13, 168.192.1-24.n.2.14, 168.192.1-24.n.2.15, 168.192.1-24.n.2.16, 264.192.1-24.n.2.1, 264.192.1-24.n.2.2, 264.192.1-24.n.2.3, 264.192.1-24.n.2.4, 264.192.1-24.n.2.5, 264.192.1-24.n.2.6, 264.192.1-24.n.2.7, 264.192.1-24.n.2.8, 264.192.1-24.n.2.9, 264.192.1-24.n.2.10, 264.192.1-24.n.2.11, 264.192.1-24.n.2.12, 264.192.1-24.n.2.13, 264.192.1-24.n.2.14, 264.192.1-24.n.2.15, 264.192.1-24.n.2.16, 312.192.1-24.n.2.1, 312.192.1-24.n.2.2, 312.192.1-24.n.2.3, 312.192.1-24.n.2.4, 312.192.1-24.n.2.5, 312.192.1-24.n.2.6, 312.192.1-24.n.2.7, 312.192.1-24.n.2.8, 312.192.1-24.n.2.9, 312.192.1-24.n.2.10, 312.192.1-24.n.2.11, 312.192.1-24.n.2.12, 312.192.1-24.n.2.13, 312.192.1-24.n.2.14, 312.192.1-24.n.2.15, 312.192.1-24.n.2.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $768$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 9x $
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^4}\cdot\frac{57348x^{2}y^{28}z^{2}+28217745630x^{2}y^{24}z^{6}+1702325241245511x^{2}y^{20}z^{10}-8119633115233342035x^{2}y^{16}z^{14}+25859339264100689920584x^{2}y^{12}z^{18}-35988833860673726883403485x^{2}y^{8}z^{22}+9178730331116150925790415541x^{2}y^{4}z^{26}-148695418365105736174136457735x^{2}z^{30}+72xy^{30}z+1060165746xy^{26}z^{5}+3078357112152xy^{22}z^{9}+559343178786328857xy^{18}z^{13}-2580026717665216890576xy^{14}z^{17}+4488406598071779348409785xy^{10}z^{21}-2719623699147462323753465100xy^{6}z^{25}+181738856485713392638390465137xy^{2}z^{29}+y^{32}+3026808y^{28}z^{4}+6663859867548y^{24}z^{8}-29627258969642526y^{20}z^{12}+129176662264973890524y^{16}z^{16}-247842484655981377266576y^{12}z^{20}+211526419194369812018721726y^{8}z^{24}-18357457744392591677917918842y^{4}z^{28}+79766443076872509863361z^{32}}{z^{2}y^{8}(x^{2}y^{20}-114453x^{2}y^{16}z^{4}-8832549420x^{2}y^{12}z^{8}-120605935328145x^{2}y^{8}z^{12}+277643203126256493x^{2}y^{4}z^{16}-13493075341822822215x^{2}z^{20}-3969xy^{18}z^{3}-11809800xy^{14}z^{7}+704459589165xy^{10}z^{11}-41131405877993172xy^{6}z^{15}+10494797169090723633xy^{2}z^{19}+54y^{20}z^{2}+8017542y^{16}z^{6}+155465624376y^{12}z^{10}+2285105998608162y^{8}z^{14}-999466727430619458y^{4}z^{18}+1853020188851841z^{22})}$

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
8.48.0.b.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.b.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.n.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.l.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.n.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.p.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.s.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.x.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.ba.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bh.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.bj.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.17.mz.1 $24$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
24.384.17.er.1 $24$ $4$ $4$ $17$ $3$ $1^{8}\cdot2^{4}$
120.192.5.cs.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ct.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.fb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.fc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.gc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.gd.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.cs.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ct.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dv.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.dw.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fb.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.fc.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.gc.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.gd.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.cs.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ct.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.dv.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.dw.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.fb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.fc.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.gc.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.gd.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.cs.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ct.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.dv.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.dw.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.fb.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.fc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.gc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.gd.2 $312$ $2$ $2$ $5$ $?$ not computed