↑

Properties

Label 24.96.1.de.4
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $24$
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 $2^{8}$
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: 24.96.1.2261

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&3\\0&23\end{bmatrix}$, $\begin{bmatrix}5&16\\0&11\end{bmatrix}$, $\begin{bmatrix}7&17\\0&1\end{bmatrix}$, $\begin{bmatrix}13&15\\0&11\end{bmatrix}$, $\begin{bmatrix}17&10\\0&5\end{bmatrix}$, $\begin{bmatrix}17&18\\0&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{24}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.de.4.1, 24.192.1-24.de.4.2, 24.192.1-24.de.4.3, 24.192.1-24.de.4.4, 24.192.1-24.de.4.5, 24.192.1-24.de.4.6, 24.192.1-24.de.4.7, 24.192.1-24.de.4.8, 24.192.1-24.de.4.9, 24.192.1-24.de.4.10, 24.192.1-24.de.4.11, 24.192.1-24.de.4.12, 24.192.1-24.de.4.13, 24.192.1-24.de.4.14, 24.192.1-24.de.4.15, 24.192.1-24.de.4.16, 24.192.1-24.de.4.17, 24.192.1-24.de.4.18, 24.192.1-24.de.4.19, 24.192.1-24.de.4.20, 24.192.1-24.de.4.21, 24.192.1-24.de.4.22, 24.192.1-24.de.4.23, 24.192.1-24.de.4.24, 48.192.1-24.de.4.1, 48.192.1-24.de.4.2, 48.192.1-24.de.4.3, 48.192.1-24.de.4.4, 48.192.1-24.de.4.5, 48.192.1-24.de.4.6, 48.192.1-24.de.4.7, 48.192.1-24.de.4.8, 48.192.1-24.de.4.9, 48.192.1-24.de.4.10, 48.192.1-24.de.4.11, 48.192.1-24.de.4.12, 48.192.1-24.de.4.13, 48.192.1-24.de.4.14, 48.192.1-24.de.4.15, 48.192.1-24.de.4.16, 120.192.1-24.de.4.1, 120.192.1-24.de.4.2, 120.192.1-24.de.4.3, 120.192.1-24.de.4.4, 120.192.1-24.de.4.5, 120.192.1-24.de.4.6, 120.192.1-24.de.4.7, 120.192.1-24.de.4.8, 120.192.1-24.de.4.9, 120.192.1-24.de.4.10, 120.192.1-24.de.4.11, 120.192.1-24.de.4.12, 120.192.1-24.de.4.13, 120.192.1-24.de.4.14, 120.192.1-24.de.4.15, 120.192.1-24.de.4.16, 120.192.1-24.de.4.17, 120.192.1-24.de.4.18, 120.192.1-24.de.4.19, 120.192.1-24.de.4.20, 120.192.1-24.de.4.21, 120.192.1-24.de.4.22, 120.192.1-24.de.4.23, 120.192.1-24.de.4.24, 168.192.1-24.de.4.1, 168.192.1-24.de.4.2, 168.192.1-24.de.4.3, 168.192.1-24.de.4.4, 168.192.1-24.de.4.5, 168.192.1-24.de.4.6, 168.192.1-24.de.4.7, 168.192.1-24.de.4.8, 168.192.1-24.de.4.9, 168.192.1-24.de.4.10, 168.192.1-24.de.4.11, 168.192.1-24.de.4.12, 168.192.1-24.de.4.13, 168.192.1-24.de.4.14, 168.192.1-24.de.4.15, 168.192.1-24.de.4.16, 168.192.1-24.de.4.17, 168.192.1-24.de.4.18, 168.192.1-24.de.4.19, 168.192.1-24.de.4.20, 168.192.1-24.de.4.21, 168.192.1-24.de.4.22, 168.192.1-24.de.4.23, 168.192.1-24.de.4.24, 240.192.1-24.de.4.1, 240.192.1-24.de.4.2, 240.192.1-24.de.4.3, 240.192.1-24.de.4.4, 240.192.1-24.de.4.5, 240.192.1-24.de.4.6, 240.192.1-24.de.4.7, 240.192.1-24.de.4.8, 240.192.1-24.de.4.9, 240.192.1-24.de.4.10, 240.192.1-24.de.4.11, 240.192.1-24.de.4.12, 240.192.1-24.de.4.13, 240.192.1-24.de.4.14, 240.192.1-24.de.4.15, 240.192.1-24.de.4.16, 264.192.1-24.de.4.1, 264.192.1-24.de.4.2, 264.192.1-24.de.4.3, 264.192.1-24.de.4.4, 264.192.1-24.de.4.5, 264.192.1-24.de.4.6, 264.192.1-24.de.4.7, 264.192.1-24.de.4.8, 264.192.1-24.de.4.9, 264.192.1-24.de.4.10, 264.192.1-24.de.4.11, 264.192.1-24.de.4.12, 264.192.1-24.de.4.13, 264.192.1-24.de.4.14, 264.192.1-24.de.4.15, 264.192.1-24.de.4.16, 264.192.1-24.de.4.17, 264.192.1-24.de.4.18, 264.192.1-24.de.4.19, 264.192.1-24.de.4.20, 264.192.1-24.de.4.21, 264.192.1-24.de.4.22, 264.192.1-24.de.4.23, 264.192.1-24.de.4.24, 312.192.1-24.de.4.1, 312.192.1-24.de.4.2, 312.192.1-24.de.4.3, 312.192.1-24.de.4.4, 312.192.1-24.de.4.5, 312.192.1-24.de.4.6, 312.192.1-24.de.4.7, 312.192.1-24.de.4.8, 312.192.1-24.de.4.9, 312.192.1-24.de.4.10, 312.192.1-24.de.4.11, 312.192.1-24.de.4.12, 312.192.1-24.de.4.13, 312.192.1-24.de.4.14, 312.192.1-24.de.4.15, 312.192.1-24.de.4.16, 312.192.1-24.de.4.17, 312.192.1-24.de.4.18, 312.192.1-24.de.4.19, 312.192.1-24.de.4.20, 312.192.1-24.de.4.21, 312.192.1-24.de.4.22, 312.192.1-24.de.4.23, 312.192.1-24.de.4.24
Cyclic 24-isogeny field degree: $1$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

$ 0 $ $=$ $ 6 x^{2} - 6 x y - 3 z^{2} - z w $
$=$ $6 x^{2} + 6 x y - 6 y^{2} - 3 z^{2} - 3 z w - w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} + 2 x^{2} y^{2} + 4 x^{2} y z - y^{2} z^{2} - 6 y z^{3} - 9 z^{4} $
 Copy content Toggle raw display

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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{188116992y^{2}z^{22}+707678208y^{2}z^{21}w-17830803456y^{2}z^{20}w^{2}-334442151936y^{2}z^{19}w^{3}-2349465025536y^{2}z^{18}w^{4}-9508915897344y^{2}z^{17}w^{5}-25835575656960y^{2}z^{16}w^{6}-50965343907840y^{2}z^{15}w^{7}-76559233605120y^{2}z^{14}w^{8}-90486907494912y^{2}z^{13}w^{9}-86210109298944y^{2}z^{12}w^{10}-67420438096896y^{2}z^{11}w^{11}-43818678117504y^{2}z^{10}w^{12}-23811367139712y^{2}z^{9}w^{13}-10809643285440y^{2}z^{8}w^{14}-4068365978880y^{2}z^{7}w^{15}-1251696402000y^{2}z^{6}w^{16}-308420506224y^{2}z^{5}w^{17}-59158469016y^{2}z^{4}w^{18}-8480854656y^{2}z^{3}w^{19}-852489036y^{2}z^{2}w^{20}-53477172y^{2}zw^{21}-1572858y^{2}w^{22}-5971968z^{24}-752467968z^{23}w-34032752640z^{22}w^{2}-649002129408z^{21}w^{3}-4875101365248z^{20}w^{4}-20707349815296z^{19}w^{5}-58501743395328z^{18}w^{6}-119968652966400z^{17}w^{7}-188401731528960z^{16}w^{8}-234791053443072z^{15}w^{9}-238035219879168z^{14}w^{10}-199822606297344z^{13}w^{11}-140636062014336z^{12}w^{12}-83664107435520z^{11}w^{13}-42248597160384z^{10}w^{14}-18116084030400z^{9}w^{15}-6571630880640z^{8}w^{16}-2000338265664z^{7}w^{17}-504268788024z^{6}w^{18}-103282499448z^{5}w^{19}-16721654868z^{4}w^{20}-2055213504z^{3}w^{21}-179831322z^{2}w^{22}-9961506zw^{23}-262145w^{24}}{wz^{3}(2z+w)^{3}(3z+w)(20736y^{2}z^{14}+214272y^{2}z^{13}w+756864y^{2}z^{12}w^{2}+1393152y^{2}z^{11}w^{3}+1664832y^{2}z^{10}w^{4}+1483968y^{2}z^{9}w^{5}+1053216y^{2}z^{8}w^{6}+603648y^{2}z^{7}w^{7}+280944y^{2}z^{6}w^{8}+107184y^{2}z^{5}w^{9}+33144y^{2}z^{4}w^{10}+7872y^{2}z^{3}w^{11}+1308y^{2}z^{2}w^{12}+132y^{2}zw^{13}+6y^{2}w^{14}+41472z^{16}-38016z^{14}w^{2}-1920z^{13}w^{3}-35392z^{12}w^{4}-114176z^{11}w^{5}-145952z^{10}w^{6}-139296z^{9}w^{7}-110928z^{8}w^{8}-69888z^{7}w^{9}-34872z^{6}w^{10}-14344z^{5}w^{11}-4828z^{4}w^{12}-1232z^{3}w^{13}-214z^{2}w^{14}-22zw^{15}-w^{16})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0.bs.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
$X_0(24)$ $24$ $2$ $2$ $1$ $0$ 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.dg.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dk.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ds.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.ew.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ey.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fe.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fg.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fz.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.ga.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gb.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gc.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gd.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.ge.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gf.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gg.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.288.9.r.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.5.ka.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kb.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kc.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kd.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ki.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kj.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kk.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kl.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kr.3 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ks.3 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.kz.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.la.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.9.bbm.2 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.192.9.bbn.2 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.bgc.1 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
48.192.9.bgd.1 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
72.288.9.s.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.ep.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ff.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ze.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbo.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bby.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdz.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bea.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beb.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bec.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bed.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bee.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bef.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beg.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zc.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ze.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zk.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zm.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbo.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbq.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbw.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bby.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdz.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bea.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beb.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bec.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bed.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bee.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bef.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beg.4 $168$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjl.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjn.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjs.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjt.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cju.4 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjv.4 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cki.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckj.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cky.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckz.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.frd.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fre.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsj.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsk.1 $240$ $2$ $2$ $9$ $?$ not computed
264.192.5.zc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ze.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zk.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zm.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbo.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbq.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbw.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bby.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdz.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bea.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beb.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bec.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bed.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bee.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bef.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beg.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ze.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zk.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zm.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbo.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbq.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbw.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bby.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdz.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bea.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beb.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bec.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bed.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bee.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bef.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beg.3 $312$ $2$ $2$ $5$ $?$ not computed