Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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: | 24.96.1.1570 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&15\\12&7\end{bmatrix}$, $\begin{bmatrix}1&18\\12&23\end{bmatrix}$, $\begin{bmatrix}7&0\\20&23\end{bmatrix}$, $\begin{bmatrix}11&15\\20&23\end{bmatrix}$, $\begin{bmatrix}19&18\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035917 |
| Contains $-I$: | no $\quad$ (see 24.48.1.iu.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
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 |
|---|---|---|---|---|---|---|
| 12.48.0-12.g.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.19 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-24.ba.1.12 | $24$ | $4$ | $4$ | $0$ | $0$ | 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.dl.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dl.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dl.3.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dl.4.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dn.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dn.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dn.3.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dn.4.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.ch.1.30 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cp.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.em.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.eo.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ez.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.fa.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ga.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gd.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gs.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gs.4.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gu.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gu.2.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gu.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gu.4.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.er.1.2 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 72.288.5-72.bo.1.10 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.dc.1.10 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.dk.1.4 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.tb.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tb.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tb.3.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tb.4.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.td.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.td.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.td.3.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.td.4.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.os.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ou.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pa.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pc.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.py.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qa.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qg.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qi.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sy.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sy.2.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sy.3.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sy.4.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ta.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ta.2.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ta.3.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ta.4.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.brm.1.45 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.sz.1.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sz.2.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sz.3.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sz.4.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tb.1.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tb.2.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tb.3.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tb.4.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.me.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mg.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mm.1.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mo.1.19 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nk.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nm.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ns.1.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nu.1.11 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qk.1.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qk.2.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qk.3.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qk.4.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qm.1.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qm.2.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qm.3.10 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qm.4.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.sz.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sz.2.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sz.3.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sz.4.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tb.1.11 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tb.2.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tb.3.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tb.4.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.me.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mg.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mm.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mo.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nk.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nm.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ns.1.17 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nu.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qk.1.19 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qk.2.18 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qk.3.3 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qk.4.2 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qm.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qm.2.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qm.3.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qm.4.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.tb.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tb.2.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tb.3.17 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tb.4.17 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.td.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.td.2.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.td.3.17 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.td.4.17 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.os.1.10 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ou.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pa.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pc.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.py.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qa.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qg.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qi.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sy.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sy.2.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sy.3.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sy.4.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ta.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ta.2.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ta.3.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ta.4.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |