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Properties

Label 24.96.1.dm.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: $24$ 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: $0$
$\Q$-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: 24.96.1.1641

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&12\\4&11\end{bmatrix}$, $\begin{bmatrix}7&18\\4&5\end{bmatrix}$, $\begin{bmatrix}11&9\\0&5\end{bmatrix}$, $\begin{bmatrix}19&18\\16&5\end{bmatrix}$, $\begin{bmatrix}23&9\\8&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035912
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dm.1.1, 24.192.1-24.dm.1.2, 24.192.1-24.dm.1.3, 24.192.1-24.dm.1.4, 24.192.1-24.dm.1.5, 24.192.1-24.dm.1.6, 24.192.1-24.dm.1.7, 24.192.1-24.dm.1.8, 24.192.1-24.dm.1.9, 24.192.1-24.dm.1.10, 24.192.1-24.dm.1.11, 24.192.1-24.dm.1.12, 24.192.1-24.dm.1.13, 24.192.1-24.dm.1.14, 24.192.1-24.dm.1.15, 24.192.1-24.dm.1.16, 120.192.1-24.dm.1.1, 120.192.1-24.dm.1.2, 120.192.1-24.dm.1.3, 120.192.1-24.dm.1.4, 120.192.1-24.dm.1.5, 120.192.1-24.dm.1.6, 120.192.1-24.dm.1.7, 120.192.1-24.dm.1.8, 120.192.1-24.dm.1.9, 120.192.1-24.dm.1.10, 120.192.1-24.dm.1.11, 120.192.1-24.dm.1.12, 120.192.1-24.dm.1.13, 120.192.1-24.dm.1.14, 120.192.1-24.dm.1.15, 120.192.1-24.dm.1.16, 168.192.1-24.dm.1.1, 168.192.1-24.dm.1.2, 168.192.1-24.dm.1.3, 168.192.1-24.dm.1.4, 168.192.1-24.dm.1.5, 168.192.1-24.dm.1.6, 168.192.1-24.dm.1.7, 168.192.1-24.dm.1.8, 168.192.1-24.dm.1.9, 168.192.1-24.dm.1.10, 168.192.1-24.dm.1.11, 168.192.1-24.dm.1.12, 168.192.1-24.dm.1.13, 168.192.1-24.dm.1.14, 168.192.1-24.dm.1.15, 168.192.1-24.dm.1.16, 264.192.1-24.dm.1.1, 264.192.1-24.dm.1.2, 264.192.1-24.dm.1.3, 264.192.1-24.dm.1.4, 264.192.1-24.dm.1.5, 264.192.1-24.dm.1.6, 264.192.1-24.dm.1.7, 264.192.1-24.dm.1.8, 264.192.1-24.dm.1.9, 264.192.1-24.dm.1.10, 264.192.1-24.dm.1.11, 264.192.1-24.dm.1.12, 264.192.1-24.dm.1.13, 264.192.1-24.dm.1.14, 264.192.1-24.dm.1.15, 264.192.1-24.dm.1.16, 312.192.1-24.dm.1.1, 312.192.1-24.dm.1.2, 312.192.1-24.dm.1.3, 312.192.1-24.dm.1.4, 312.192.1-24.dm.1.5, 312.192.1-24.dm.1.6, 312.192.1-24.dm.1.7, 312.192.1-24.dm.1.8, 312.192.1-24.dm.1.9, 312.192.1-24.dm.1.10, 312.192.1-24.dm.1.11, 312.192.1-24.dm.1.12, 312.192.1-24.dm.1.13, 312.192.1-24.dm.1.14, 312.192.1-24.dm.1.15, 312.192.1-24.dm.1.16
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.d

Models

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

$ 0 $ $=$ $ x z + x w - y z $
$=$ $2 x^{2} - 8 x y + 2 y^{2} - 2 z^{2} - 2 z w - 5 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} + 6 x^{3} z + 4 x^{2} y^{2} + 11 x^{2} z^{2} + 4 x y^{2} z + 12 x z^{3} - 2 y^{2} z^{2} + 5 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^3}\cdot\frac{18791216660726415360xy^{23}+118456965621111324672xy^{21}w^{2}+321835909508166057984xy^{19}w^{4}+481312934940575268864xy^{17}w^{6}+458768694210003468288xy^{15}w^{8}+196484138525994909696xy^{13}w^{10}+385024309484162383872xy^{11}w^{12}-1391995095650810044416xy^{9}w^{14}+6980938208633202444288xy^{7}w^{16}-34986915363353231174400xy^{5}w^{18}+177643282597970865401088xy^{3}w^{20}-911688844317745358199312xyw^{22}-5035091329038680064y^{24}-15466777279790579712y^{22}w^{2}-2900890059754438656y^{20}w^{4}+88106305586385125376y^{18}w^{6}+118568360280373788672y^{16}w^{8}+387797400757455814656y^{14}w^{10}-926828544415940591616y^{12}w^{12}+5037598441350634438656y^{10}w^{14}-24718635338195840295168y^{8}w^{16}+124080294481134659504640y^{6}w^{18}-630454270366456837537824y^{4}w^{20}+3237525326641636717240320y^{2}w^{22}-3358429036052480z^{24}-76014597485248512z^{23}w-892024595606913024z^{22}w^{2}-7203894891416428544z^{21}w^{3}-44822351290703020032z^{20}w^{4}-228178101906882613248z^{19}w^{5}-986180705060305375232z^{18}w^{6}-3709366926836640196608z^{17}w^{7}-12357541104444938631168z^{16}w^{8}-36938396400404680017920z^{15}w^{9}-100043603224766902192128z^{14}w^{10}-247359293411362649490432z^{13}w^{11}-561561555216774325643264z^{12}w^{12}-1175688286234244460341760z^{11}w^{13}-2277031221333468153484800z^{10}w^{14}-4087864948000720090882304z^{9}w^{15}-6807399226733874033753984z^{8}w^{16}-10509030712451070677005632z^{7}w^{17}-14994790316210046225070016z^{6}w^{18}-19673635638135641179550496z^{5}w^{19}-23409761730890626510269312z^{4}w^{20}-24848332175770346204145296z^{3}w^{21}-21595478007000266202941328z^{2}w^{22}-14626170345313479617545416zw^{23}-5449509520499202873920351w^{24}}{w^{6}(z-w)(z+2w)(2z+w)^{2}(z^{2}+zw+w^{2})^{3}(2z^{2}+2zw+5w^{2})^{4}}$

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.4 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bs.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iv.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.cw.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.dj.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ek.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.el.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eu.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ey.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fu.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fv.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.u.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.ba.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ex.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fn.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bai.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bak.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bay.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bba.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcw.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdk.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdm.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bai.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bak.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bay.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bba.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcu.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcw.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdk.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdm.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bai.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bak.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bay.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bba.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcu.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcw.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdk.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdm.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bai.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bak.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bay.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bba.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcu.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcw.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdk.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdm.4 $312$ $2$ $2$ $5$ $?$ not computed