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Properties

Label 24.96.1.cp.3
Level $24$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ 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: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-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: 24.96.1.1332

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\18&23\end{bmatrix}$, $\begin{bmatrix}5&10\\18&1\end{bmatrix}$, $\begin{bmatrix}11&8\\6&1\end{bmatrix}$, $\begin{bmatrix}13&12\\6&1\end{bmatrix}$, $\begin{bmatrix}17&18\\6&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.335742
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cp.3.1, 24.192.1-24.cp.3.2, 24.192.1-24.cp.3.3, 24.192.1-24.cp.3.4, 24.192.1-24.cp.3.5, 24.192.1-24.cp.3.6, 24.192.1-24.cp.3.7, 24.192.1-24.cp.3.8, 24.192.1-24.cp.3.9, 24.192.1-24.cp.3.10, 24.192.1-24.cp.3.11, 24.192.1-24.cp.3.12, 24.192.1-24.cp.3.13, 24.192.1-24.cp.3.14, 24.192.1-24.cp.3.15, 24.192.1-24.cp.3.16, 120.192.1-24.cp.3.1, 120.192.1-24.cp.3.2, 120.192.1-24.cp.3.3, 120.192.1-24.cp.3.4, 120.192.1-24.cp.3.5, 120.192.1-24.cp.3.6, 120.192.1-24.cp.3.7, 120.192.1-24.cp.3.8, 120.192.1-24.cp.3.9, 120.192.1-24.cp.3.10, 120.192.1-24.cp.3.11, 120.192.1-24.cp.3.12, 120.192.1-24.cp.3.13, 120.192.1-24.cp.3.14, 120.192.1-24.cp.3.15, 120.192.1-24.cp.3.16, 168.192.1-24.cp.3.1, 168.192.1-24.cp.3.2, 168.192.1-24.cp.3.3, 168.192.1-24.cp.3.4, 168.192.1-24.cp.3.5, 168.192.1-24.cp.3.6, 168.192.1-24.cp.3.7, 168.192.1-24.cp.3.8, 168.192.1-24.cp.3.9, 168.192.1-24.cp.3.10, 168.192.1-24.cp.3.11, 168.192.1-24.cp.3.12, 168.192.1-24.cp.3.13, 168.192.1-24.cp.3.14, 168.192.1-24.cp.3.15, 168.192.1-24.cp.3.16, 264.192.1-24.cp.3.1, 264.192.1-24.cp.3.2, 264.192.1-24.cp.3.3, 264.192.1-24.cp.3.4, 264.192.1-24.cp.3.5, 264.192.1-24.cp.3.6, 264.192.1-24.cp.3.7, 264.192.1-24.cp.3.8, 264.192.1-24.cp.3.9, 264.192.1-24.cp.3.10, 264.192.1-24.cp.3.11, 264.192.1-24.cp.3.12, 264.192.1-24.cp.3.13, 264.192.1-24.cp.3.14, 264.192.1-24.cp.3.15, 264.192.1-24.cp.3.16, 312.192.1-24.cp.3.1, 312.192.1-24.cp.3.2, 312.192.1-24.cp.3.3, 312.192.1-24.cp.3.4, 312.192.1-24.cp.3.5, 312.192.1-24.cp.3.6, 312.192.1-24.cp.3.7, 312.192.1-24.cp.3.8, 312.192.1-24.cp.3.9, 312.192.1-24.cp.3.10, 312.192.1-24.cp.3.11, 312.192.1-24.cp.3.12, 312.192.1-24.cp.3.13, 312.192.1-24.cp.3.14, 312.192.1-24.cp.3.15, 312.192.1-24.cp.3.16
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

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

Models

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

$ 0 $ $=$ $ x z + y w $
$=$ $3 x^{2} - y^{2} - 6 z^{2} - 6 w^{2}$
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Singular plane model Singular plane model

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

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{y^{24}+72y^{22}w^{2}-216y^{20}w^{4}-14688y^{18}w^{6}-11664y^{16}w^{8}+15116544y^{14}w^{10}-1172931840y^{12}w^{12}+86164300800y^{10}w^{14}-6285474111744y^{8}w^{16}+459378751776768y^{6}w^{18}-33782007862941696y^{4}w^{20}+2502639516206702592y^{2}w^{22}-2173796352z^{24}-78256668672z^{22}w^{2}-1445598461952z^{20}w^{4}-18469433769984z^{18}w^{6}-184568568053760z^{16}w^{8}-1541138674876416z^{14}w^{10}-11147674491813888z^{12}w^{12}-70951216019668992z^{10}w^{14}-395780979633979392z^{8}w^{16}-1867902077142171648z^{6}w^{18}-6538577334941122560z^{4}w^{20}-5005279023706275840z^{2}w^{22}+2176782336w^{24}}{w^{4}(y^{12}w^{8}-72y^{10}w^{10}+4536y^{8}w^{12}-296352y^{6}w^{14}+20042640y^{4}w^{16}-1391468544y^{2}w^{18}+64z^{20}+2304z^{18}w^{2}+41344z^{16}w^{4}+499968z^{14}w^{6}+4598272z^{12}w^{8}+34078464z^{10}w^{10}+207909504z^{8}w^{12}+1025443584z^{6}w^{14}+3630412224z^{4}w^{16}+2782937088z^{2}w^{18})}$

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
12.48.0.a.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.o.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.bz.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.bq.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.br.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bs.2 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.bt.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bv.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bw.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.by.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.bz.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.288.9.e.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.h.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.p.4 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.bb.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.kl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.km.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ko.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kp.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ks.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kt.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kv.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kw.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.kl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.km.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ko.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kp.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ks.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kt.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kv.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kw.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.kl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.km.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ko.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kp.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ks.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kt.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kv.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kw.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.kl.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.km.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ko.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kp.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ks.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kt.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kv.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kw.4 $312$ $2$ $2$ $5$ $?$ not computed