Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $72$ | ||
| 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: | 24.96.1.2157 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&18\\20&17\end{bmatrix}$, $\begin{bmatrix}5&21\\20&5\end{bmatrix}$, $\begin{bmatrix}13&9\\4&23\end{bmatrix}$, $\begin{bmatrix}17&12\\16&17\end{bmatrix}$, $\begin{bmatrix}19&21\\16&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035913 |
| Contains $-I$: | no $\quad$ (see 24.48.1.it.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: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 39x - 70 $ |
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 |
|---|
| $(0:1:0)$, $(7:0:1)$, $(-2:0:1)$, $(-5:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{48x^{2}y^{14}+250290x^{2}y^{12}z^{2}+704021544x^{2}y^{10}z^{4}+1036544079285x^{2}y^{8}z^{6}+792049398959880x^{2}y^{6}z^{8}+328014454394251449x^{2}y^{4}z^{10}+61839042257985762780x^{2}y^{2}z^{12}+4145763953672617760757x^{2}z^{14}+1092xy^{14}z+4500360xy^{12}z^{3}+9035067807xy^{10}z^{5}+11068238211714xy^{8}z^{7}+7707764288002176xy^{6}z^{9}+2801618589805099920xy^{4}z^{11}+472047366323568417561xy^{2}z^{13}+29056820286475034636310xz^{15}+y^{16}+15600y^{14}z^{2}+58606740y^{12}z^{4}+92100693600y^{10}z^{6}+80931580839972y^{8}z^{8}+42434572546725408y^{6}z^{10}+10661845166918803242y^{4}z^{12}+1158729840347955398568y^{2}z^{14}+41640003747391110588801z^{16}}{z^{2}y^{2}(x^{2}y^{10}+9828x^{2}y^{8}z^{2}+6271587x^{2}y^{6}z^{4}+873137880x^{2}y^{4}z^{6}-68024448x^{2}y^{2}z^{8}+918330048x^{2}z^{10}+40xy^{10}z+112779xy^{8}z^{3}+52069554xy^{6}z^{5}+6093069480xy^{4}z^{7}+365631408xy^{2}z^{9}-4591650240xz^{11}+742y^{10}z^{2}+784512y^{8}z^{4}+175578192y^{6}z^{6}+8713427904y^{4}z^{8}+697250592y^{2}z^{10}-12856620672z^{12})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.24.0-24.z.1.6 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.di.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.di.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.di.3.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.di.4.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dk.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dk.2.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dk.3.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dk.4.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.ch.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cq.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.dt.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.du.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fd.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fe.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fs.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fv.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.gp.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gp.2.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gp.3.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gp.4.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gr.1.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gr.2.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gr.3.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gr.4.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.eu.1.10 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 72.288.5-72.bn.1.3 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.db.1.3 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.dj.1.8 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.si.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.si.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.si.3.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.si.4.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sk.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sk.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sk.3.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sk.4.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ol.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.on.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.op.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.or.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pr.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pt.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pv.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.px.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sf.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sf.2.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sf.3.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sf.4.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sh.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sh.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sh.3.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sh.4.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.brd.1.11 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.sg.1.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sg.2.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sg.3.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sg.4.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.si.1.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.si.2.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.si.3.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.si.4.24 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.lx.1.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.lz.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mb.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.md.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nd.1.30 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nf.1.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nh.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nj.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pr.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pr.2.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pr.3.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pr.4.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pt.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pt.2.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pt.3.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pt.4.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.sg.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sg.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sg.3.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sg.4.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.si.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.si.2.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.si.3.14 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.si.4.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.lx.1.7 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.lz.1.18 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mb.1.31 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.md.1.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nd.1.31 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nf.1.27 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nh.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nj.1.18 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pr.1.6 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pr.2.7 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pr.3.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pr.4.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pt.1.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pt.2.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pt.3.8 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pt.4.8 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.si.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.si.2.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.si.3.23 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.si.4.23 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sk.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sk.2.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sk.3.23 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sk.4.23 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ol.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.on.1.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.op.1.10 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.or.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pr.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pt.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pv.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.px.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sf.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sf.2.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sf.3.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sf.4.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sh.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sh.2.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sh.3.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sh.4.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |