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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $24$ 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 $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-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: 24.96.1.1628

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&21\\12&17\end{bmatrix}$, $\begin{bmatrix}1&21\\16&19\end{bmatrix}$, $\begin{bmatrix}13&3\\0&5\end{bmatrix}$, $\begin{bmatrix}17&0\\0&13\end{bmatrix}$, $\begin{bmatrix}23&18\\12&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035916
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dh.2.1, 24.192.1-24.dh.2.2, 24.192.1-24.dh.2.3, 24.192.1-24.dh.2.4, 24.192.1-24.dh.2.5, 24.192.1-24.dh.2.6, 24.192.1-24.dh.2.7, 24.192.1-24.dh.2.8, 24.192.1-24.dh.2.9, 24.192.1-24.dh.2.10, 24.192.1-24.dh.2.11, 24.192.1-24.dh.2.12, 24.192.1-24.dh.2.13, 24.192.1-24.dh.2.14, 24.192.1-24.dh.2.15, 24.192.1-24.dh.2.16, 120.192.1-24.dh.2.1, 120.192.1-24.dh.2.2, 120.192.1-24.dh.2.3, 120.192.1-24.dh.2.4, 120.192.1-24.dh.2.5, 120.192.1-24.dh.2.6, 120.192.1-24.dh.2.7, 120.192.1-24.dh.2.8, 120.192.1-24.dh.2.9, 120.192.1-24.dh.2.10, 120.192.1-24.dh.2.11, 120.192.1-24.dh.2.12, 120.192.1-24.dh.2.13, 120.192.1-24.dh.2.14, 120.192.1-24.dh.2.15, 120.192.1-24.dh.2.16, 168.192.1-24.dh.2.1, 168.192.1-24.dh.2.2, 168.192.1-24.dh.2.3, 168.192.1-24.dh.2.4, 168.192.1-24.dh.2.5, 168.192.1-24.dh.2.6, 168.192.1-24.dh.2.7, 168.192.1-24.dh.2.8, 168.192.1-24.dh.2.9, 168.192.1-24.dh.2.10, 168.192.1-24.dh.2.11, 168.192.1-24.dh.2.12, 168.192.1-24.dh.2.13, 168.192.1-24.dh.2.14, 168.192.1-24.dh.2.15, 168.192.1-24.dh.2.16, 264.192.1-24.dh.2.1, 264.192.1-24.dh.2.2, 264.192.1-24.dh.2.3, 264.192.1-24.dh.2.4, 264.192.1-24.dh.2.5, 264.192.1-24.dh.2.6, 264.192.1-24.dh.2.7, 264.192.1-24.dh.2.8, 264.192.1-24.dh.2.9, 264.192.1-24.dh.2.10, 264.192.1-24.dh.2.11, 264.192.1-24.dh.2.12, 264.192.1-24.dh.2.13, 264.192.1-24.dh.2.14, 264.192.1-24.dh.2.15, 264.192.1-24.dh.2.16, 312.192.1-24.dh.2.1, 312.192.1-24.dh.2.2, 312.192.1-24.dh.2.3, 312.192.1-24.dh.2.4, 312.192.1-24.dh.2.5, 312.192.1-24.dh.2.6, 312.192.1-24.dh.2.7, 312.192.1-24.dh.2.8, 312.192.1-24.dh.2.9, 312.192.1-24.dh.2.10, 312.192.1-24.dh.2.11, 312.192.1-24.dh.2.12, 312.192.1-24.dh.2.13, 312.192.1-24.dh.2.14, 312.192.1-24.dh.2.15, 312.192.1-24.dh.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
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 w + y z $
$=$ $6 x^{2} + 6 y^{2} - z^{2} - 4 z w - w^{2}$
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Singular plane model Singular plane model

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

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{35664401793024xy^{21}w^{2}+77272870551552xy^{19}w^{4}-37975983390720xy^{17}w^{6}-233304628396032xy^{15}w^{8}-175698547310592xy^{13}w^{10}+110744939003904xy^{11}w^{12}+244977198170112xy^{9}w^{14}+110697800269824xy^{7}w^{16}-58633797500928xy^{5}w^{18}-93382932824064xy^{3}w^{20}-39260945448960xyw^{22}-8916100448256y^{24}-17832200896512y^{22}w^{2}+49038552465408y^{20}w^{4}+141501816373248y^{18}w^{6}+58621984505856y^{16}w^{8}-146134152511488y^{14}w^{10}-189585618370560y^{12}w^{12}-36044314509312y^{10}w^{14}+92701935599616y^{8}w^{16}+84550669959168y^{6}w^{18}+17616144531456y^{4}w^{20}-19638868279296y^{2}w^{22}-z^{24}-24z^{23}w-252z^{22}w^{2}-776z^{21}w^{3}+9822z^{20}w^{4}+126168z^{19}w^{5}+488692z^{18}w^{6}-1383288z^{17}w^{7}-22454895z^{16}w^{8}-81777392z^{15}w^{9}+71489544z^{14}w^{10}+1609016496z^{13}w^{11}+4931555364z^{12}w^{12}-630741840z^{11}w^{13}-40596482040z^{10}w^{14}-82734732016z^{9}w^{15}+86900301201z^{8}w^{16}+577378116744z^{7}w^{17}+417484731636z^{6}w^{18}-1964318266152z^{5}w^{19}-3912681642402z^{4}w^{20}+3225428163832z^{3}w^{21}+16534349020932z^{2}w^{22}+12829736730600zw^{23}+2330010939391w^{24}}{w^{3}z^{3}(z-w)^{2}(z+w)^{6}(z^{2}+zw+w^{2})(z^{2}+4zw+w^{2})^{4}}$

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.c.3 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bs.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.is.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.cw.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.di.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dw.1 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.dz.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fc.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fg.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fm.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fn.3 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.288.9.z.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
72.288.9.v.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.es.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fi.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zt.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zz.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bab.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcf.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcn.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zr.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zt.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zz.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bab.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcd.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcf.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcn.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.zr.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zt.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zz.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bab.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcf.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcn.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zr.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zt.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zz.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bab.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcd.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcf.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcn.1 $312$ $2$ $2$ $5$ $?$ not computed