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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $72$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ 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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1938

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&21\\0&13\end{bmatrix}$, $\begin{bmatrix}13&9\\0&11\end{bmatrix}$, $\begin{bmatrix}13&11\\0&17\end{bmatrix}$, $\begin{bmatrix}17&0\\12&5\end{bmatrix}$, $\begin{bmatrix}19&6\\0&7\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035915
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cz.1.1, 24.192.1-24.cz.1.2, 24.192.1-24.cz.1.3, 24.192.1-24.cz.1.4, 24.192.1-24.cz.1.5, 24.192.1-24.cz.1.6, 24.192.1-24.cz.1.7, 24.192.1-24.cz.1.8, 24.192.1-24.cz.1.9, 24.192.1-24.cz.1.10, 24.192.1-24.cz.1.11, 24.192.1-24.cz.1.12, 24.192.1-24.cz.1.13, 24.192.1-24.cz.1.14, 24.192.1-24.cz.1.15, 24.192.1-24.cz.1.16, 120.192.1-24.cz.1.1, 120.192.1-24.cz.1.2, 120.192.1-24.cz.1.3, 120.192.1-24.cz.1.4, 120.192.1-24.cz.1.5, 120.192.1-24.cz.1.6, 120.192.1-24.cz.1.7, 120.192.1-24.cz.1.8, 120.192.1-24.cz.1.9, 120.192.1-24.cz.1.10, 120.192.1-24.cz.1.11, 120.192.1-24.cz.1.12, 120.192.1-24.cz.1.13, 120.192.1-24.cz.1.14, 120.192.1-24.cz.1.15, 120.192.1-24.cz.1.16, 168.192.1-24.cz.1.1, 168.192.1-24.cz.1.2, 168.192.1-24.cz.1.3, 168.192.1-24.cz.1.4, 168.192.1-24.cz.1.5, 168.192.1-24.cz.1.6, 168.192.1-24.cz.1.7, 168.192.1-24.cz.1.8, 168.192.1-24.cz.1.9, 168.192.1-24.cz.1.10, 168.192.1-24.cz.1.11, 168.192.1-24.cz.1.12, 168.192.1-24.cz.1.13, 168.192.1-24.cz.1.14, 168.192.1-24.cz.1.15, 168.192.1-24.cz.1.16, 264.192.1-24.cz.1.1, 264.192.1-24.cz.1.2, 264.192.1-24.cz.1.3, 264.192.1-24.cz.1.4, 264.192.1-24.cz.1.5, 264.192.1-24.cz.1.6, 264.192.1-24.cz.1.7, 264.192.1-24.cz.1.8, 264.192.1-24.cz.1.9, 264.192.1-24.cz.1.10, 264.192.1-24.cz.1.11, 264.192.1-24.cz.1.12, 264.192.1-24.cz.1.13, 264.192.1-24.cz.1.14, 264.192.1-24.cz.1.15, 264.192.1-24.cz.1.16, 312.192.1-24.cz.1.1, 312.192.1-24.cz.1.2, 312.192.1-24.cz.1.3, 312.192.1-24.cz.1.4, 312.192.1-24.cz.1.5, 312.192.1-24.cz.1.6, 312.192.1-24.cz.1.7, 312.192.1-24.cz.1.8, 312.192.1-24.cz.1.9, 312.192.1-24.cz.1.10, 312.192.1-24.cz.1.11, 312.192.1-24.cz.1.12, 312.192.1-24.cz.1.13, 312.192.1-24.cz.1.14, 312.192.1-24.cz.1.15, 312.192.1-24.cz.1.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

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

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

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3}\cdot\frac{(3z^{2}-w^{2})^{3}(7164612y^{2}z^{16}+20890224y^{2}z^{14}w^{2}+17880912y^{2}z^{12}w^{4}+92110608y^{2}z^{10}w^{6}+13516632y^{2}z^{8}w^{8}+10234512y^{2}z^{6}w^{10}+220752y^{2}z^{4}w^{12}+28656y^{2}z^{2}w^{14}+1092y^{2}w^{16}+4782969z^{18}+14348907z^{16}w^{2}+21423852z^{14}w^{4}+19522620z^{12}w^{6}+11773350z^{10}w^{8}+3445578z^{8}w^{10}+1093068z^{6}w^{12}-37044z^{4}w^{14}-1239z^{2}w^{16}-61w^{18})}{w^{2}z^{2}(3z^{2}+w^{2})^{4}(1458y^{2}z^{10}+3402y^{2}z^{8}w^{2}-4212y^{2}z^{6}w^{4}+1404y^{2}z^{4}w^{6}-126y^{2}z^{2}w^{8}-6y^{2}w^{10}-4941z^{8}w^{4}-864z^{6}w^{6}-378z^{4}w^{8}-24z^{2}w^{10}-w^{12})}$

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
12.48.1.k.1 $12$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bu.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.2 $24$ $2$ $2$ $0$ $0$ full Jacobian

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.et.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.et.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ew.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ew.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fk.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fk.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fl.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fl.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.bj.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
72.288.9.l.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.eg.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.eh.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.wb.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wb.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wf.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wr.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wr.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wv.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.wb.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wb.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wr.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wr.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wv.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wv.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.wb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wb.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wf.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wf.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wr.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wr.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wv.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wv.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.wb.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wb.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wf.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wf.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wr.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wr.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wv.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wv.4 $312$ $2$ $2$ $5$ $?$ not computed