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Properties

Label 24.96.1.dd.3
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: $192$
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: 24.96.1.1640

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&9\\0&1\end{bmatrix}$, $\begin{bmatrix}17&0\\16&11\end{bmatrix}$, $\begin{bmatrix}17&9\\4&11\end{bmatrix}$, $\begin{bmatrix}17&12\\0&19\end{bmatrix}$, $\begin{bmatrix}23&12\\20&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dd.3.1, 24.192.1-24.dd.3.2, 24.192.1-24.dd.3.3, 24.192.1-24.dd.3.4, 24.192.1-24.dd.3.5, 24.192.1-24.dd.3.6, 24.192.1-24.dd.3.7, 24.192.1-24.dd.3.8, 24.192.1-24.dd.3.9, 24.192.1-24.dd.3.10, 24.192.1-24.dd.3.11, 24.192.1-24.dd.3.12, 24.192.1-24.dd.3.13, 24.192.1-24.dd.3.14, 24.192.1-24.dd.3.15, 24.192.1-24.dd.3.16, 120.192.1-24.dd.3.1, 120.192.1-24.dd.3.2, 120.192.1-24.dd.3.3, 120.192.1-24.dd.3.4, 120.192.1-24.dd.3.5, 120.192.1-24.dd.3.6, 120.192.1-24.dd.3.7, 120.192.1-24.dd.3.8, 120.192.1-24.dd.3.9, 120.192.1-24.dd.3.10, 120.192.1-24.dd.3.11, 120.192.1-24.dd.3.12, 120.192.1-24.dd.3.13, 120.192.1-24.dd.3.14, 120.192.1-24.dd.3.15, 120.192.1-24.dd.3.16, 168.192.1-24.dd.3.1, 168.192.1-24.dd.3.2, 168.192.1-24.dd.3.3, 168.192.1-24.dd.3.4, 168.192.1-24.dd.3.5, 168.192.1-24.dd.3.6, 168.192.1-24.dd.3.7, 168.192.1-24.dd.3.8, 168.192.1-24.dd.3.9, 168.192.1-24.dd.3.10, 168.192.1-24.dd.3.11, 168.192.1-24.dd.3.12, 168.192.1-24.dd.3.13, 168.192.1-24.dd.3.14, 168.192.1-24.dd.3.15, 168.192.1-24.dd.3.16, 264.192.1-24.dd.3.1, 264.192.1-24.dd.3.2, 264.192.1-24.dd.3.3, 264.192.1-24.dd.3.4, 264.192.1-24.dd.3.5, 264.192.1-24.dd.3.6, 264.192.1-24.dd.3.7, 264.192.1-24.dd.3.8, 264.192.1-24.dd.3.9, 264.192.1-24.dd.3.10, 264.192.1-24.dd.3.11, 264.192.1-24.dd.3.12, 264.192.1-24.dd.3.13, 264.192.1-24.dd.3.14, 264.192.1-24.dd.3.15, 264.192.1-24.dd.3.16, 312.192.1-24.dd.3.1, 312.192.1-24.dd.3.2, 312.192.1-24.dd.3.3, 312.192.1-24.dd.3.4, 312.192.1-24.dd.3.5, 312.192.1-24.dd.3.6, 312.192.1-24.dd.3.7, 312.192.1-24.dd.3.8, 312.192.1-24.dd.3.9, 312.192.1-24.dd.3.10, 312.192.1-24.dd.3.11, 312.192.1-24.dd.3.12, 312.192.1-24.dd.3.13, 312.192.1-24.dd.3.14, 312.192.1-24.dd.3.15, 312.192.1-24.dd.3.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

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

$ 0 $ $=$ $ y^{2} - 4 y z - y w + z^{2} - z w $
$=$ $6 x^{2} + 2 y w + 2 z w + w^{2}$
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Singular plane model Singular plane model

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

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^3}\cdot\frac{18791216660726415360yz^{23}+168446375618823585792yz^{22}w+702526400696539938816yz^{21}w^{2}+1810803822673224794112yz^{20}w^{3}+3231908463969375879168yz^{19}w^{4}+4241238201739794972672yz^{18}w^{5}+4241597876602384416768yz^{17}w^{6}+3306630154817864466432yz^{16}w^{7}+2038466905844028014592yz^{15}w^{8}+1002434822239735775232yz^{14}w^{9}+394944571160710742016yz^{13}w^{10}+124756362293923676160yz^{12}w^{11}+31513665833700360192yz^{11}w^{12}+6327287669947023360yz^{10}w^{13}+999782853118119936yz^{9}w^{14}+122504919990865920yz^{8}w^{15}+11394583901816832yz^{7}w^{16}+779901290860032yz^{6}w^{17}+37477886194944yz^{5}w^{18}+1172331484416yz^{4}w^{19}+20885921280yz^{3}w^{20}+164689728yz^{2}w^{21}+573000yzw^{22}+728yw^{23}-5035091329038680064z^{24}-38257007649166983168z^{23}w-129298007575643553792z^{22}w^{2}-251017913279509954560z^{21}w^{3}-293222525392820109312z^{20}w^{4}-170725073647388590080z^{19}w^{5}+54700887333157208064z^{18}w^{6}+227503229656528060416z^{17}w^{7}+259941431780154998784z^{16}w^{8}+189522469512577548288z^{15}w^{9}+100080022188975980544z^{14}w^{10}+39977204527698149376z^{13}w^{11}+12290227132987883520z^{12}w^{12}+2921762327736729600z^{11}w^{13}+535216358607814656z^{10}w^{14}+74736540896514048z^{9}w^{15}+7807000225174272z^{8}w^{16}+592285748590080z^{7}w^{17}+31155391383552z^{6}w^{18}+1052801756928z^{5}w^{19}+19918232640z^{4}w^{20}+161277504z^{3}w^{21}+568704z^{2}w^{22}+728zw^{23}+w^{24}}{w^{12}z(1347840yz^{10}+5704128yz^{9}w+10419408yz^{8}w^{2}+10787088yz^{7}w^{3}+6988872yz^{6}w^{4}+2952046yz^{5}w^{5}+820827yz^{4}w^{6}+147933yz^{3}w^{7}+16497yz^{2}w^{8}+1026yzw^{9}+27yw^{10}-361152z^{11}-1035072z^{10}w-898560z^{9}w^{2}+197328z^{8}w^{3}+926691z^{7}w^{4}+758422z^{6}w^{5}+321056z^{5}w^{6}+79083z^{4}w^{7}+11313z^{3}w^{8}+864z^{2}w^{9}+27zw^{10})}$

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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.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iq.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.de.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dh.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dq.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dt.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eu.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ex.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fc.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ff.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.q.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.r.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.eo.4 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fe.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zj.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbv.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbx.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zb.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zd.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zj.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zl.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbn.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbp.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbv.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbx.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.zb.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zd.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zj.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbn.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbp.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbv.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbx.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zb.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zd.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zj.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbn.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbp.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbv.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbx.1 $312$ $2$ $2$ $5$ $?$ not computed