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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2235

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&18\\0&13\end{bmatrix}$, $\begin{bmatrix}5&18\\12&7\end{bmatrix}$, $\begin{bmatrix}17&21\\16&23\end{bmatrix}$, $\begin{bmatrix}23&0\\20&5\end{bmatrix}$, $\begin{bmatrix}23&3\\0&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035859
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.ds.2.1, 24.192.1-24.ds.2.2, 24.192.1-24.ds.2.3, 24.192.1-24.ds.2.4, 24.192.1-24.ds.2.5, 24.192.1-24.ds.2.6, 24.192.1-24.ds.2.7, 24.192.1-24.ds.2.8, 24.192.1-24.ds.2.9, 24.192.1-24.ds.2.10, 24.192.1-24.ds.2.11, 24.192.1-24.ds.2.12, 24.192.1-24.ds.2.13, 24.192.1-24.ds.2.14, 24.192.1-24.ds.2.15, 24.192.1-24.ds.2.16, 120.192.1-24.ds.2.1, 120.192.1-24.ds.2.2, 120.192.1-24.ds.2.3, 120.192.1-24.ds.2.4, 120.192.1-24.ds.2.5, 120.192.1-24.ds.2.6, 120.192.1-24.ds.2.7, 120.192.1-24.ds.2.8, 120.192.1-24.ds.2.9, 120.192.1-24.ds.2.10, 120.192.1-24.ds.2.11, 120.192.1-24.ds.2.12, 120.192.1-24.ds.2.13, 120.192.1-24.ds.2.14, 120.192.1-24.ds.2.15, 120.192.1-24.ds.2.16, 168.192.1-24.ds.2.1, 168.192.1-24.ds.2.2, 168.192.1-24.ds.2.3, 168.192.1-24.ds.2.4, 168.192.1-24.ds.2.5, 168.192.1-24.ds.2.6, 168.192.1-24.ds.2.7, 168.192.1-24.ds.2.8, 168.192.1-24.ds.2.9, 168.192.1-24.ds.2.10, 168.192.1-24.ds.2.11, 168.192.1-24.ds.2.12, 168.192.1-24.ds.2.13, 168.192.1-24.ds.2.14, 168.192.1-24.ds.2.15, 168.192.1-24.ds.2.16, 264.192.1-24.ds.2.1, 264.192.1-24.ds.2.2, 264.192.1-24.ds.2.3, 264.192.1-24.ds.2.4, 264.192.1-24.ds.2.5, 264.192.1-24.ds.2.6, 264.192.1-24.ds.2.7, 264.192.1-24.ds.2.8, 264.192.1-24.ds.2.9, 264.192.1-24.ds.2.10, 264.192.1-24.ds.2.11, 264.192.1-24.ds.2.12, 264.192.1-24.ds.2.13, 264.192.1-24.ds.2.14, 264.192.1-24.ds.2.15, 264.192.1-24.ds.2.16, 312.192.1-24.ds.2.1, 312.192.1-24.ds.2.2, 312.192.1-24.ds.2.3, 312.192.1-24.ds.2.4, 312.192.1-24.ds.2.5, 312.192.1-24.ds.2.6, 312.192.1-24.ds.2.7, 312.192.1-24.ds.2.8, 312.192.1-24.ds.2.9, 312.192.1-24.ds.2.10, 312.192.1-24.ds.2.11, 312.192.1-24.ds.2.12, 312.192.1-24.ds.2.13, 312.192.1-24.ds.2.14, 312.192.1-24.ds.2.15, 312.192.1-24.ds.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} - 2 x z + w^{2} $
$=$ $2 x^{2} - 4 x y + 2 y^{2} - 2 y z + z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} - 6 x^{3} y + 2 x^{2} y^{2} + 4 x^{2} z^{2} - 4 x y z^{2} + 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{728xz^{23}-95488xz^{21}w^{2}+4542720xz^{19}w^{4}-93649920xz^{17}w^{6}+809693184xz^{15}w^{8}-3694362624xz^{13}w^{10}+9898164224xz^{11}w^{12}-16171663360xz^{9}w^{14}+15944122368xz^{7}w^{16}-8900313088xz^{5}w^{18}+2399141888xz^{3}w^{20}-201326592xzw^{22}+z^{24}-376z^{22}w^{2}+47616z^{20}w^{4}-2247680z^{18}w^{6}+45712128z^{16}w^{8}-382537728z^{14}w^{10}+1666793472z^{12}w^{12}-4200595456z^{10}w^{14}+6319964160z^{8}w^{16}-5546442752z^{6}w^{18}+2587885568z^{4}w^{20}-503316480z^{2}w^{22}+16777216w^{24}}{w^{2}z^{12}(z^{2}-2w^{2})(486xz^{7}-1944xz^{5}w^{2}+2336xz^{3}w^{4}-768xzw^{6}-243z^{6}w^{2}+850z^{4}w^{4}-800z^{2}w^{6}+128w^{8})}$

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
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.ix.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.dd.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.do.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ep.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.es.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fl.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fq.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ft.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fy.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.bb.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.bc.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.fd.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ft.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bau.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baw.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbk.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbm.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdg.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdi.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdw.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdy.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bau.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.baw.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbk.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbm.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdg.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdi.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdw.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdy.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bau.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.baw.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbk.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbm.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdg.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdi.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdw.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdy.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bau.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.baw.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbk.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbm.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdg.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdi.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdw.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdy.1 $312$ $2$ $2$ $5$ $?$ not computed