Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| 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: | 24.96.1.97 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&9\\12&23\end{bmatrix}$, $\begin{bmatrix}19&15\\20&23\end{bmatrix}$, $\begin{bmatrix}19&18\\4&7\end{bmatrix}$, $\begin{bmatrix}23&6\\8&13\end{bmatrix}$, $\begin{bmatrix}23&18\\4&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035859 |
| Contains $-I$: | no $\quad$ (see 24.48.1.ix.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: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 17x + 15 $ |
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 |
|---|
| $(0:1:0)$, $(-5:0:1)$, $(1:0:1)$, $(3:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4}\cdot\frac{32x^{2}y^{14}-49440x^{2}y^{12}z^{2}+41204736x^{2}y^{10}z^{4}-17975260160x^{2}y^{8}z^{6}+4069732843520x^{2}y^{6}z^{8}-499381945565184x^{2}y^{4}z^{10}+27895158454353920x^{2}y^{2}z^{12}-554111127877844992x^{2}z^{14}-464xy^{14}z+559680xy^{12}z^{3}-325064448xy^{10}z^{5}+115976615936xy^{8}z^{7}-23689694150656xy^{6}z^{9}+2510607340929024xy^{4}z^{11}-123361399543955456xy^{2}z^{13}+2219694395765030912xz^{15}-y^{16}+4464y^{14}z^{2}-4953120y^{12}z^{4}+2282817280y^{10}z^{6}-583111623680y^{8}z^{8}+88557304872960y^{6}z^{10}-6321869422133248y^{4}z^{12}+188088919423713280y^{2}z^{14}-1672083105113964544z^{16}}{z^{2}y^{2}(x^{2}y^{10}-2912x^{2}y^{8}z^{2}+550592x^{2}y^{6}z^{4}-22712320x^{2}y^{4}z^{6}-524288x^{2}y^{2}z^{8}-2097152x^{2}z^{10}-26xy^{10}z+20336xy^{8}z^{3}-2680448xy^{6}z^{5}+90521600xy^{4}z^{7}-2228224xy^{2}z^{9}-8388608xz^{11}+321y^{10}z^{2}-96208y^{8}z^{4}+5896128y^{6}z^{6}-68038656y^{4}z^{8}+1703936y^{2}z^{10}+10485760z^{12})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.p.1.7 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.p.1.7 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.19 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.dq.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dq.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dq.3.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.dq.4.9 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ds.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ds.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ds.3.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ds.4.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.ck.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cr.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.er.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.es.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ft.1.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fv.1.3 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.gb.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gd.1.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gx.1.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gx.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gx.3.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gx.4.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gz.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gz.2.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gz.3.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gz.4.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.288.5-24.fd.1.5 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 72.288.5-72.br.1.7 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.df.1.7 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.dn.1.13 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.tg.1.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tg.2.19 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tg.3.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tg.4.21 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ti.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ti.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ti.3.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ti.4.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ox.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oz.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pf.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ph.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qd.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qf.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ql.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.qn.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.td.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.td.2.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.td.3.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.td.4.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tf.1.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tf.2.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tf.3.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tf.4.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.brp.1.53 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.te.1.26 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.te.2.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.te.3.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.te.4.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tg.1.26 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tg.2.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tg.3.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.tg.4.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.mj.1.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ml.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mr.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mt.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.np.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nr.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nx.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nz.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qp.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qp.2.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qp.3.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qp.4.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qr.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qr.2.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qr.3.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.qr.4.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.te.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.te.2.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.te.3.18 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.te.4.17 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tg.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tg.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tg.3.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.tg.4.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.mj.1.5 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ml.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mr.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mt.1.23 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.np.1.3 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nr.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nx.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nz.1.23 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qp.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qp.2.3 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qp.3.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qp.4.3 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qr.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qr.2.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qr.3.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.qr.4.17 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.tg.1.21 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tg.2.19 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tg.3.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.tg.4.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ti.1.21 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ti.2.21 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ti.3.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ti.4.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ox.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.oz.1.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pf.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ph.1.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qd.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qf.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ql.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.qn.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.td.1.21 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.td.2.21 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.td.3.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.td.4.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tf.1.21 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tf.2.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tf.3.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tf.4.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |