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Properties

Label 24.96.1.dl.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: $144$
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 $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2266

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&12\\20&23\end{bmatrix}$, $\begin{bmatrix}7&15\\4&13\end{bmatrix}$, $\begin{bmatrix}11&0\\16&11\end{bmatrix}$, $\begin{bmatrix}23&3\\20&23\end{bmatrix}$, $\begin{bmatrix}23&21\\20&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035917
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dl.2.1, 24.192.1-24.dl.2.2, 24.192.1-24.dl.2.3, 24.192.1-24.dl.2.4, 24.192.1-24.dl.2.5, 24.192.1-24.dl.2.6, 24.192.1-24.dl.2.7, 24.192.1-24.dl.2.8, 24.192.1-24.dl.2.9, 24.192.1-24.dl.2.10, 24.192.1-24.dl.2.11, 24.192.1-24.dl.2.12, 24.192.1-24.dl.2.13, 24.192.1-24.dl.2.14, 24.192.1-24.dl.2.15, 24.192.1-24.dl.2.16, 120.192.1-24.dl.2.1, 120.192.1-24.dl.2.2, 120.192.1-24.dl.2.3, 120.192.1-24.dl.2.4, 120.192.1-24.dl.2.5, 120.192.1-24.dl.2.6, 120.192.1-24.dl.2.7, 120.192.1-24.dl.2.8, 120.192.1-24.dl.2.9, 120.192.1-24.dl.2.10, 120.192.1-24.dl.2.11, 120.192.1-24.dl.2.12, 120.192.1-24.dl.2.13, 120.192.1-24.dl.2.14, 120.192.1-24.dl.2.15, 120.192.1-24.dl.2.16, 168.192.1-24.dl.2.1, 168.192.1-24.dl.2.2, 168.192.1-24.dl.2.3, 168.192.1-24.dl.2.4, 168.192.1-24.dl.2.5, 168.192.1-24.dl.2.6, 168.192.1-24.dl.2.7, 168.192.1-24.dl.2.8, 168.192.1-24.dl.2.9, 168.192.1-24.dl.2.10, 168.192.1-24.dl.2.11, 168.192.1-24.dl.2.12, 168.192.1-24.dl.2.13, 168.192.1-24.dl.2.14, 168.192.1-24.dl.2.15, 168.192.1-24.dl.2.16, 264.192.1-24.dl.2.1, 264.192.1-24.dl.2.2, 264.192.1-24.dl.2.3, 264.192.1-24.dl.2.4, 264.192.1-24.dl.2.5, 264.192.1-24.dl.2.6, 264.192.1-24.dl.2.7, 264.192.1-24.dl.2.8, 264.192.1-24.dl.2.9, 264.192.1-24.dl.2.10, 264.192.1-24.dl.2.11, 264.192.1-24.dl.2.12, 264.192.1-24.dl.2.13, 264.192.1-24.dl.2.14, 264.192.1-24.dl.2.15, 264.192.1-24.dl.2.16, 312.192.1-24.dl.2.1, 312.192.1-24.dl.2.2, 312.192.1-24.dl.2.3, 312.192.1-24.dl.2.4, 312.192.1-24.dl.2.5, 312.192.1-24.dl.2.6, 312.192.1-24.dl.2.7, 312.192.1-24.dl.2.8, 312.192.1-24.dl.2.9, 312.192.1-24.dl.2.10, 312.192.1-24.dl.2.11, 312.192.1-24.dl.2.12, 312.192.1-24.dl.2.13, 312.192.1-24.dl.2.14, 312.192.1-24.dl.2.15, 312.192.1-24.dl.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{4}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 144.2.a.b

Models

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

$ 0 $ $=$ $ 6 x^{2} + 4 x y - 2 y^{2} + z^{2} $
$=$ $6 x^{2} - 10 x y + 2 y^{2} - z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 6 x^{2} y^{2} - 4 x^{2} z^{2} - 12 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3}\cdot\frac{(3z^{2}-2w^{2})^{3}(9552816y^{2}z^{16}-50948352y^{2}z^{14}w^{2}+112510944y^{2}z^{12}w^{4}-133031808y^{2}z^{10}w^{6}+87391872y^{2}z^{8}w^{8}-26735616y^{2}z^{6}w^{10}+1128960y^{2}z^{4}w^{12}-104448y^{2}z^{2}w^{14}-186368y^{2}w^{16}+6383853z^{18}-44671662z^{16}w^{2}+135576504z^{14}w^{4}-231417648z^{12}w^{6}+237536064z^{10}w^{8}-141730560z^{8}w^{10}+41014656z^{6}w^{12}-2436864z^{4}w^{14}+86784z^{2}w^{16}-10752w^{18})}{w^{8}z^{2}(3z^{2}-4w^{2})(486y^{2}z^{10}-1620y^{2}z^{8}w^{2}+1836y^{2}z^{6}w^{4}-792y^{2}z^{4}w^{6}+48y^{2}z^{2}w^{8}+32y^{2}w^{10}-243z^{12}+648z^{10}w^{2}-567z^{8}w^{4}+288z^{6}w^{6}-75z^{4}w^{8}-60z^{2}w^{10}-48w^{12})}$

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
24.48.0.bs.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iu.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.cy.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.di.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ei.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.el.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ew.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ex.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fs.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fw.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.v.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
72.288.9.z.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ew.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fm.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bah.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baj.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bax.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bct.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdl.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bah.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.baj.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bax.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.baz.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bct.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcv.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdj.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdl.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bah.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.baj.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bax.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.baz.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bct.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcv.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdj.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdl.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bah.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.baj.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bax.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.baz.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bct.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcv.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdj.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdl.1 $312$ $2$ $2$ $5$ $?$ not computed