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Properties

Label 24.96.1.co.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: 24.96.1.1885

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&8\\18&19\end{bmatrix}$, $\begin{bmatrix}7&20\\6&1\end{bmatrix}$, $\begin{bmatrix}11&12\\6&7\end{bmatrix}$, $\begin{bmatrix}17&2\\0&13\end{bmatrix}$, $\begin{bmatrix}17&20\\18&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335743
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.co.1.1, 24.192.1-24.co.1.2, 24.192.1-24.co.1.3, 24.192.1-24.co.1.4, 24.192.1-24.co.1.5, 24.192.1-24.co.1.6, 24.192.1-24.co.1.7, 24.192.1-24.co.1.8, 24.192.1-24.co.1.9, 24.192.1-24.co.1.10, 24.192.1-24.co.1.11, 24.192.1-24.co.1.12, 24.192.1-24.co.1.13, 24.192.1-24.co.1.14, 24.192.1-24.co.1.15, 24.192.1-24.co.1.16, 120.192.1-24.co.1.1, 120.192.1-24.co.1.2, 120.192.1-24.co.1.3, 120.192.1-24.co.1.4, 120.192.1-24.co.1.5, 120.192.1-24.co.1.6, 120.192.1-24.co.1.7, 120.192.1-24.co.1.8, 120.192.1-24.co.1.9, 120.192.1-24.co.1.10, 120.192.1-24.co.1.11, 120.192.1-24.co.1.12, 120.192.1-24.co.1.13, 120.192.1-24.co.1.14, 120.192.1-24.co.1.15, 120.192.1-24.co.1.16, 168.192.1-24.co.1.1, 168.192.1-24.co.1.2, 168.192.1-24.co.1.3, 168.192.1-24.co.1.4, 168.192.1-24.co.1.5, 168.192.1-24.co.1.6, 168.192.1-24.co.1.7, 168.192.1-24.co.1.8, 168.192.1-24.co.1.9, 168.192.1-24.co.1.10, 168.192.1-24.co.1.11, 168.192.1-24.co.1.12, 168.192.1-24.co.1.13, 168.192.1-24.co.1.14, 168.192.1-24.co.1.15, 168.192.1-24.co.1.16, 264.192.1-24.co.1.1, 264.192.1-24.co.1.2, 264.192.1-24.co.1.3, 264.192.1-24.co.1.4, 264.192.1-24.co.1.5, 264.192.1-24.co.1.6, 264.192.1-24.co.1.7, 264.192.1-24.co.1.8, 264.192.1-24.co.1.9, 264.192.1-24.co.1.10, 264.192.1-24.co.1.11, 264.192.1-24.co.1.12, 264.192.1-24.co.1.13, 264.192.1-24.co.1.14, 264.192.1-24.co.1.15, 264.192.1-24.co.1.16, 312.192.1-24.co.1.1, 312.192.1-24.co.1.2, 312.192.1-24.co.1.3, 312.192.1-24.co.1.4, 312.192.1-24.co.1.5, 312.192.1-24.co.1.6, 312.192.1-24.co.1.7, 312.192.1-24.co.1.8, 312.192.1-24.co.1.9, 312.192.1-24.co.1.10, 312.192.1-24.co.1.11, 312.192.1-24.co.1.12, 312.192.1-24.co.1.13, 312.192.1-24.co.1.14, 312.192.1-24.co.1.15, 312.192.1-24.co.1.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
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 $ $=$ $ 4 x y + 2 y^{2} + z^{2} $
$=$ $6 x^{2} + 4 x y - 4 y^{2} - 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} + 6 x^{2} y^{2} + 4 x^{2} 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{3^2}\cdot\frac{(9z^{4}+3z^{2}w^{2}+w^{4})(7164612y^{2}z^{18}+10746918y^{2}z^{16}w^{2}+6219828y^{2}z^{14}w^{4}+1683990y^{2}z^{12}w^{6}+135594y^{2}z^{10}w^{8}-45198y^{2}z^{8}w^{10}-62370y^{2}z^{6}w^{12}-25596y^{2}z^{4}w^{14}-4914y^{2}z^{2}w^{16}-364y^{2}w^{18}-1200663z^{20}-1404054z^{18}w^{2}-579555z^{16}w^{4}-114453z^{14}w^{6}-70956z^{12}w^{8}-57672z^{10}w^{10}-46602z^{8}w^{12}-24606z^{6}w^{14}-7518z^{4}w^{16}-1215z^{2}w^{18}-81w^{20})}{w^{4}z^{4}(3z^{2}+w^{2})^{2}(1944y^{2}z^{10}+1620y^{2}z^{8}w^{2}+216y^{2}z^{6}w^{4}-72y^{2}z^{4}w^{6}-60y^{2}z^{2}w^{8}-8y^{2}w^{10}+972z^{12}+1134z^{10}w^{2}+405z^{8}w^{4}+18z^{6}w^{6}+3z^{4}w^{8})}$

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
12.48.1.d.1 $12$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.o.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.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.bp.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bp.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bt.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bt.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bu.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bu.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ca.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ca.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.f.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
72.288.9.g.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.o.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ba.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.kk.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kk.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kn.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kn.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kr.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kr.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ku.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ku.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.kk.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kk.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kn.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kn.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kr.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kr.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ku.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ku.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.kk.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kk.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kn.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kn.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kr.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kr.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ku.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ku.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.kk.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kk.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kn.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kn.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kr.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kr.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ku.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ku.4 $312$ $2$ $2$ $5$ $?$ not computed