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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $1^{4}\cdot4^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.661

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\8&17\end{bmatrix}$, $\begin{bmatrix}5&14\\8&7\end{bmatrix}$, $\begin{bmatrix}7&4\\8&1\end{bmatrix}$, $\begin{bmatrix}23&12\\16&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1089047
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cd.1.1, 24.192.1-24.cd.1.2, 24.192.1-24.cd.1.3, 24.192.1-24.cd.1.4, 24.192.1-24.cd.1.5, 24.192.1-24.cd.1.6, 24.192.1-24.cd.1.7, 24.192.1-24.cd.1.8, 48.192.1-24.cd.1.1, 48.192.1-24.cd.1.2, 48.192.1-24.cd.1.3, 48.192.1-24.cd.1.4, 48.192.1-24.cd.1.5, 48.192.1-24.cd.1.6, 48.192.1-24.cd.1.7, 48.192.1-24.cd.1.8, 48.192.1-24.cd.1.9, 48.192.1-24.cd.1.10, 48.192.1-24.cd.1.11, 48.192.1-24.cd.1.12, 120.192.1-24.cd.1.1, 120.192.1-24.cd.1.2, 120.192.1-24.cd.1.3, 120.192.1-24.cd.1.4, 120.192.1-24.cd.1.5, 120.192.1-24.cd.1.6, 120.192.1-24.cd.1.7, 120.192.1-24.cd.1.8, 168.192.1-24.cd.1.1, 168.192.1-24.cd.1.2, 168.192.1-24.cd.1.3, 168.192.1-24.cd.1.4, 168.192.1-24.cd.1.5, 168.192.1-24.cd.1.6, 168.192.1-24.cd.1.7, 168.192.1-24.cd.1.8, 240.192.1-24.cd.1.1, 240.192.1-24.cd.1.2, 240.192.1-24.cd.1.3, 240.192.1-24.cd.1.4, 240.192.1-24.cd.1.5, 240.192.1-24.cd.1.6, 240.192.1-24.cd.1.7, 240.192.1-24.cd.1.8, 240.192.1-24.cd.1.9, 240.192.1-24.cd.1.10, 240.192.1-24.cd.1.11, 240.192.1-24.cd.1.12, 264.192.1-24.cd.1.1, 264.192.1-24.cd.1.2, 264.192.1-24.cd.1.3, 264.192.1-24.cd.1.4, 264.192.1-24.cd.1.5, 264.192.1-24.cd.1.6, 264.192.1-24.cd.1.7, 264.192.1-24.cd.1.8, 312.192.1-24.cd.1.1, 312.192.1-24.cd.1.2, 312.192.1-24.cd.1.3, 312.192.1-24.cd.1.4, 312.192.1-24.cd.1.5, 312.192.1-24.cd.1.6, 312.192.1-24.cd.1.7, 312.192.1-24.cd.1.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 9x $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:0:1)$, $(0:1:0)$, $(-3:0:1)$, $(3:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^2}\cdot\frac{14112468x^{2}y^{28}z^{2}+178127386413570x^{2}y^{24}z^{6}+6962494991310477351x^{2}y^{20}z^{10}+18041756667105025196595x^{2}y^{16}z^{14}+6231353476666506100422264x^{2}y^{12}z^{18}+583688060519941241389549485x^{2}y^{8}z^{22}+18357451653402196690396589541x^{2}y^{4}z^{26}+148695418365105736174136457735x^{2}z^{30}+6408xy^{30}z+2130754467186xy^{26}z^{5}+320818616388124488xy^{22}z^{9}+2052468590721326777817xy^{18}z^{13}+1518782811917983115962176xy^{14}z^{17}+234621485135854096045521465xy^{10}z^{21}+11898354747565875333902679900xy^{6}z^{25}+181738856485713392638390465137xy^{2}z^{29}+y^{32}+11567427912y^{28}z^{4}+8380519155108828y^{24}z^{8}+105003536964696697806y^{20}z^{12}+112831932610271871984924y^{16}z^{16}+20147482624549642972232976y^{12}z^{20}+1118092516920807220810928286y^{8}z^{24}+18357536526064766366816549322y^{4}z^{28}+79766443076872509863361z^{32}}{zy^{4}(279x^{2}y^{24}z-6547878x^{2}y^{20}z^{5}+928402980714x^{2}y^{16}z^{9}+6384365000350218x^{2}y^{12}z^{13}-35480743464960964647x^{2}y^{8}z^{17}-51490437158783705678505x^{2}y^{4}z^{21}-1416469339858957679723175x^{2}z^{25}-xy^{26}+1283040xy^{22}z^{4}+28807290846xy^{18}z^{8}-557233088056524xy^{14}z^{12}+1462069079984772117xy^{10}z^{16}-9823100501945895691032xy^{6}z^{20}-1416470840805310649714385xy^{2}z^{24}-30780y^{24}z^{3}-1575191124y^{20}z^{7}-25763936032431y^{16}z^{11}+189348669283883868y^{12}z^{15}-623704633846772918040y^{8}z^{19}-139897639381645192050054y^{4}z^{23}-12157665459056928801z^{27})}$

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
8.48.0.l.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.i.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.j.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bb.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.be.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bf.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.bu.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
24.288.17.blf.2 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.od.2 $24$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
48.192.5.bq.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.cb.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.dj.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dk.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dn.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.do.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dr.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ds.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dv.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.dw.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ef.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ek.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
240.192.5.sx.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.tf.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.uv.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.uw.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vi.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vx.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.vy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.xf.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.xn.1 $240$ $2$ $2$ $5$ $?$ not computed