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Properties

Label 24.96.1.dg.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

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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: $16$ (of which $4$ are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{4}\cdot2^{6}$
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: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1595

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&14\\0&17\end{bmatrix}$, $\begin{bmatrix}1&16\\0&7\end{bmatrix}$, $\begin{bmatrix}1&18\\0&13\end{bmatrix}$, $\begin{bmatrix}11&4\\0&1\end{bmatrix}$, $\begin{bmatrix}11&7\\0&13\end{bmatrix}$, $\begin{bmatrix}13&7\\0&5\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{24}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dg.2.1, 24.192.1-24.dg.2.2, 24.192.1-24.dg.2.3, 24.192.1-24.dg.2.4, 24.192.1-24.dg.2.5, 24.192.1-24.dg.2.6, 24.192.1-24.dg.2.7, 24.192.1-24.dg.2.8, 24.192.1-24.dg.2.9, 24.192.1-24.dg.2.10, 24.192.1-24.dg.2.11, 24.192.1-24.dg.2.12, 24.192.1-24.dg.2.13, 24.192.1-24.dg.2.14, 24.192.1-24.dg.2.15, 24.192.1-24.dg.2.16, 24.192.1-24.dg.2.17, 24.192.1-24.dg.2.18, 24.192.1-24.dg.2.19, 24.192.1-24.dg.2.20, 24.192.1-24.dg.2.21, 24.192.1-24.dg.2.22, 24.192.1-24.dg.2.23, 24.192.1-24.dg.2.24, 48.192.1-24.dg.2.1, 48.192.1-24.dg.2.2, 48.192.1-24.dg.2.3, 48.192.1-24.dg.2.4, 48.192.1-24.dg.2.5, 48.192.1-24.dg.2.6, 48.192.1-24.dg.2.7, 48.192.1-24.dg.2.8, 48.192.1-24.dg.2.9, 48.192.1-24.dg.2.10, 48.192.1-24.dg.2.11, 48.192.1-24.dg.2.12, 48.192.1-24.dg.2.13, 48.192.1-24.dg.2.14, 48.192.1-24.dg.2.15, 48.192.1-24.dg.2.16, 120.192.1-24.dg.2.1, 120.192.1-24.dg.2.2, 120.192.1-24.dg.2.3, 120.192.1-24.dg.2.4, 120.192.1-24.dg.2.5, 120.192.1-24.dg.2.6, 120.192.1-24.dg.2.7, 120.192.1-24.dg.2.8, 120.192.1-24.dg.2.9, 120.192.1-24.dg.2.10, 120.192.1-24.dg.2.11, 120.192.1-24.dg.2.12, 120.192.1-24.dg.2.13, 120.192.1-24.dg.2.14, 120.192.1-24.dg.2.15, 120.192.1-24.dg.2.16, 120.192.1-24.dg.2.17, 120.192.1-24.dg.2.18, 120.192.1-24.dg.2.19, 120.192.1-24.dg.2.20, 120.192.1-24.dg.2.21, 120.192.1-24.dg.2.22, 120.192.1-24.dg.2.23, 120.192.1-24.dg.2.24, 168.192.1-24.dg.2.1, 168.192.1-24.dg.2.2, 168.192.1-24.dg.2.3, 168.192.1-24.dg.2.4, 168.192.1-24.dg.2.5, 168.192.1-24.dg.2.6, 168.192.1-24.dg.2.7, 168.192.1-24.dg.2.8, 168.192.1-24.dg.2.9, 168.192.1-24.dg.2.10, 168.192.1-24.dg.2.11, 168.192.1-24.dg.2.12, 168.192.1-24.dg.2.13, 168.192.1-24.dg.2.14, 168.192.1-24.dg.2.15, 168.192.1-24.dg.2.16, 168.192.1-24.dg.2.17, 168.192.1-24.dg.2.18, 168.192.1-24.dg.2.19, 168.192.1-24.dg.2.20, 168.192.1-24.dg.2.21, 168.192.1-24.dg.2.22, 168.192.1-24.dg.2.23, 168.192.1-24.dg.2.24, 240.192.1-24.dg.2.1, 240.192.1-24.dg.2.2, 240.192.1-24.dg.2.3, 240.192.1-24.dg.2.4, 240.192.1-24.dg.2.5, 240.192.1-24.dg.2.6, 240.192.1-24.dg.2.7, 240.192.1-24.dg.2.8, 240.192.1-24.dg.2.9, 240.192.1-24.dg.2.10, 240.192.1-24.dg.2.11, 240.192.1-24.dg.2.12, 240.192.1-24.dg.2.13, 240.192.1-24.dg.2.14, 240.192.1-24.dg.2.15, 240.192.1-24.dg.2.16, 264.192.1-24.dg.2.1, 264.192.1-24.dg.2.2, 264.192.1-24.dg.2.3, 264.192.1-24.dg.2.4, 264.192.1-24.dg.2.5, 264.192.1-24.dg.2.6, 264.192.1-24.dg.2.7, 264.192.1-24.dg.2.8, 264.192.1-24.dg.2.9, 264.192.1-24.dg.2.10, 264.192.1-24.dg.2.11, 264.192.1-24.dg.2.12, 264.192.1-24.dg.2.13, 264.192.1-24.dg.2.14, 264.192.1-24.dg.2.15, 264.192.1-24.dg.2.16, 264.192.1-24.dg.2.17, 264.192.1-24.dg.2.18, 264.192.1-24.dg.2.19, 264.192.1-24.dg.2.20, 264.192.1-24.dg.2.21, 264.192.1-24.dg.2.22, 264.192.1-24.dg.2.23, 264.192.1-24.dg.2.24, 312.192.1-24.dg.2.1, 312.192.1-24.dg.2.2, 312.192.1-24.dg.2.3, 312.192.1-24.dg.2.4, 312.192.1-24.dg.2.5, 312.192.1-24.dg.2.6, 312.192.1-24.dg.2.7, 312.192.1-24.dg.2.8, 312.192.1-24.dg.2.9, 312.192.1-24.dg.2.10, 312.192.1-24.dg.2.11, 312.192.1-24.dg.2.12, 312.192.1-24.dg.2.13, 312.192.1-24.dg.2.14, 312.192.1-24.dg.2.15, 312.192.1-24.dg.2.16, 312.192.1-24.dg.2.17, 312.192.1-24.dg.2.18, 312.192.1-24.dg.2.19, 312.192.1-24.dg.2.20, 312.192.1-24.dg.2.21, 312.192.1-24.dg.2.22, 312.192.1-24.dg.2.23, 312.192.1-24.dg.2.24
Cyclic 24-isogeny field degree: $1$
Cyclic 24-torsion field degree: $4$
Full 24-torsion field degree: $768$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} + x $
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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
$(0:1:0)$, $(0:0:1)$, $(1:1:1)$, $(1:-1:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{8x^{2}y^{30}+72x^{2}y^{28}z^{2}-3000x^{2}y^{26}z^{4}+11896x^{2}y^{24}z^{6}-11064x^{2}y^{22}z^{8}-45640x^{2}y^{20}z^{10}+188328x^{2}y^{18}z^{12}-370008x^{2}y^{16}z^{14}+447192x^{2}y^{14}z^{16}-316936x^{2}y^{12}z^{18}+78552x^{2}y^{10}z^{20}+85512x^{2}y^{8}z^{22}-104104x^{2}y^{6}z^{24}+48648x^{2}y^{4}z^{26}-10184x^{2}y^{2}z^{28}+728x^{2}z^{30}-8xy^{30}z+648xy^{28}z^{3}+168xy^{26}z^{5}-18760xy^{24}z^{7}+72792xy^{22}z^{9}-144568xy^{20}z^{11}+127368xy^{18}z^{13}+72504xy^{16}z^{15}-389016xy^{14}z^{17}+599896xy^{12}z^{19}-543240xy^{10}z^{21}+301800xy^{8}z^{23}-84152xy^{6}z^{25}+216xy^{4}z^{27}+5080xy^{2}z^{29}-728xz^{31}-y^{32}-96y^{30}z^{2}-312y^{28}z^{4}+7696y^{26}z^{6}-28204y^{24}z^{8}+48256y^{22}z^{10}-23256y^{20}z^{12}-103728y^{18}z^{14}+284058y^{16}z^{16}-389728y^{14}z^{18}+334680y^{12}z^{20}-178896y^{10}z^{22}+44564y^{8}z^{24}+5440y^{6}z^{26}-5320y^{4}z^{28}+752y^{2}z^{30}-z^{32}}{z^{8}y^{8}(y-z)^{3}(y+z)^{3}(30x^{2}y^{6}z^{2}-82x^{2}y^{4}z^{4}+36x^{2}y^{2}z^{6}-12xy^{8}z+36xy^{6}z^{3}+37xy^{4}z^{5}-54xy^{2}z^{7}+9xz^{9}+y^{10}-9y^{8}z^{2}-28y^{6}z^{4}+45y^{4}z^{6}-9y^{2}z^{8})}$

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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ 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.df.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.do.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dr.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.du.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ev.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fa.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fd.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fi.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.gh.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gi.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gj.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
$X_{\pm1}(24)$ $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gl.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gm.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gn.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.go.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.288.9.p.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.5.ke.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kf.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kg.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kh.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.km.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kn.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ko.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kp.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kq.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.kt.4 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.ky.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.lb.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.9.bbl.1 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.192.9.bbo.1 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.bgb.2 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
48.192.9.bge.2 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
72.288.9.q.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.er.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fh.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zg.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zi.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zo.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbs.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbu.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bca.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcc.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bei.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bej.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bek.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bel.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bem.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ben.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beo.1 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zg.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zi.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zo.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zq.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbs.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbu.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bca.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcc.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beh.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bei.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bej.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bek.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bel.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bem.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ben.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beo.2 $168$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjo.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjp.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjw.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjx.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjz.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckk.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cla.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.clb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.frf.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.frg.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsl.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsm.2 $240$ $2$ $2$ $9$ $?$ not computed
264.192.5.zg.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zi.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zo.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zq.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbs.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbu.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bca.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcc.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beh.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bei.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bej.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bek.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bel.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bem.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ben.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beo.2 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zg.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zi.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zo.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zq.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbs.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbu.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bca.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beh.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bei.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bej.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bek.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bel.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bem.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ben.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beo.1 $312$ $2$ $2$ $5$ $?$ not computed