Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot6\cdot8\cdot24$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | 24G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1579 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\12&13\end{bmatrix}$, $\begin{bmatrix}7&12\\8&13\end{bmatrix}$, $\begin{bmatrix}7&21\\12&5\end{bmatrix}$, $\begin{bmatrix}11&12\\8&17\end{bmatrix}$, $\begin{bmatrix}19&6\\12&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035912 |
| Contains $-I$: | no $\quad$ (see 24.48.1.iv.1 for the level structure with $-I$) |
| 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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 156x - 560 $ |
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 |
|---|
| $(0:1:0)$, $(14:0:1)$, $(-4:0:1)$, $(-10:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4\cdot3^4}\cdot\frac{96x^{2}y^{14}+4004640x^{2}y^{12}z^{2}+90114757632x^{2}y^{10}z^{4}+1061421137187840x^{2}y^{8}z^{6}+6488468676279336960x^{2}y^{6}z^{8}+21496755283181662961664x^{2}y^{4}z^{10}+32421467787354839596400640x^{2}y^{2}z^{12}+17388594333944875364414128128x^{2}z^{14}+4368xy^{14}z+144011520xy^{12}z^{3}+2312977358592xy^{10}z^{5}+22667751857590272xy^{8}z^{7}+126284010094627651584xy^{6}z^{9}+367213751802934056714240xy^{4}z^{11}+494977539190102077012443136xy^{2}z^{13}+243746275109686767350427156480xz^{15}+y^{16}+124800y^{14}z^{2}+3750831360y^{12}z^{4}+47155555123200y^{10}z^{6}+331495755120525312y^{8}z^{8}+1390496073211098169344y^{6}z^{10}+2794938739436762757070848y^{4}z^{12}+2430032602145395360017678336y^{2}z^{14}+698603337110790098828201558016z^{16}}{z^{2}y^{2}(x^{2}y^{10}+78624x^{2}y^{8}z^{2}+401381568x^{2}y^{6}z^{4}+447046594560x^{2}y^{4}z^{6}-278628139008x^{2}y^{2}z^{8}+30091839012864x^{2}z^{10}+80xy^{10}z+1804464xy^{8}z^{3}+6664902912xy^{6}z^{5}+6239303147520xy^{4}z^{7}+2995252494336xy^{2}z^{9}-300918390128640xz^{11}+2968y^{10}z^{2}+25104384y^{8}z^{4}+44948017152y^{6}z^{6}+17845100347392y^{4}z^{8}+11423753699328y^{2}z^{10}-1685142984720384z^{12})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.g.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-24.bb.1.10 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.dm.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dm.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dm.3.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dm.4.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.do.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.do.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.do.3.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.do.4.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.cg.1.37 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.cq.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.en.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.eo.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ey.1.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fb.1.6 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.gb.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gc.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gt.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gt.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gt.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gt.4.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gv.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gv.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gv.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gv.4.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.eq.1.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 72.288.5-72.bp.1.5 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.dd.1.5 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.dl.1.3 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.tc.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tc.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tc.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tc.4.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.te.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.te.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.te.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.te.4.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ot.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ov.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pb.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pd.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pz.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qb.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qh.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qj.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sz.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sz.2.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sz.3.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sz.4.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tb.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tb.2.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tb.3.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tb.4.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.brn.1.37 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.ta.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ta.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ta.3.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ta.4.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tc.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tc.2.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tc.3.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tc.4.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.mf.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mh.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mn.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mp.1.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nl.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nn.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nt.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nv.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ql.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ql.2.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ql.3.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ql.4.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qn.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qn.2.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qn.3.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qn.4.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.ta.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ta.2.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ta.3.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ta.4.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tc.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tc.2.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tc.3.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tc.4.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.mf.1.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mh.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mn.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mp.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nl.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nn.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nt.1.18 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nv.1.5 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ql.1.17 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ql.2.17 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ql.3.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ql.4.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qn.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qn.2.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qn.3.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qn.4.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.tc.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tc.2.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tc.3.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tc.4.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.te.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.te.2.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.te.3.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.te.4.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ot.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ov.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pb.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pd.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pz.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qb.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qh.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qj.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sz.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sz.2.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sz.3.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sz.4.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tb.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tb.2.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tb.3.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tb.4.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |