Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.11 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\6&1\end{bmatrix}$, $\begin{bmatrix}17&0\\14&19\end{bmatrix}$, $\begin{bmatrix}19&1\\10&23\end{bmatrix}$, $\begin{bmatrix}19&3\\10&17\end{bmatrix}$, $\begin{bmatrix}23&18\\0&5\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 156x + 560 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^6}\cdot\frac{36x^{2}y^{10}+28512x^{2}y^{8}z^{2}-409826304x^{2}y^{6}z^{4}-1280310810624x^{2}y^{4}z^{6}+13560944718200832x^{2}y^{2}z^{8}-18932127234638413824x^{2}z^{10}+252xy^{10}z-72576xy^{8}z^{3}+6256569600xy^{6}z^{5}+42859393007616xy^{4}z^{7}-230957502683922432xy^{2}z^{9}+265452280319874170880xz^{11}+y^{12}+144y^{10}z^{2}+35385984y^{8}z^{4}+15190447104y^{6}z^{6}-790571790323712y^{4}z^{8}+1743857282426535936y^{2}z^{10}-760894090152386494464z^{12}}{z^{4}y^{6}(12x^{2}-204xz-y^{2}+624z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.m.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.j.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.1.ca.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.1.cb.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.1.cd.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.1.ce.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.3.br.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.da.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.gi.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.gl.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.lv.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.ly.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.mq.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qp.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.qq.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qs.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.5.a.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.108.5.w.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.72.1.ny.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.nz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ob.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.elw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.elx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.emd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eme.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eoa.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eob.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eoh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eoi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eps.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.brc.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bww.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.fs.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ft.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.fv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.fw.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.eae.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eaf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eal.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eam.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eci.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eea.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bbw.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.1.fo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fp.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fs.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.eae.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eaf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eal.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eam.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eci.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eea.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.fs.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ft.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.fv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.fw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.eae.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eaf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eal.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eam.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eci.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eea.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |