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: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.98 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\0&23\end{bmatrix}$, $\begin{bmatrix}7&1\\14&1\end{bmatrix}$, $\begin{bmatrix}15&13\\22&21\end{bmatrix}$, $\begin{bmatrix}17&10\\20&17\end{bmatrix}$, $\begin{bmatrix}19&0\\0&23\end{bmatrix}$, $\begin{bmatrix}19&12\\12&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} + z w $ |
| $=$ | $6 y^{2} - 4 z^{2} + 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + x^{2} z^{2} - 6 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(2z^{3}+w^{3})^{3}}{w^{3}z^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.l.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.a.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.i.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.3.b.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.en.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.hl.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hp.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.se.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.sg.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ss.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.su.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ws.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.wt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.xa.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xb.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.xm.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.xn.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xp.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.xq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xr.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xv.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.xy.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.xz.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bbr.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bbt.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.bfj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.bfl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.5.dl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.dn.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.jd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.jf.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.108.5.bp.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.324.21.bg.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
| 120.72.3.erg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eri.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eru.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.erw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.etk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.etm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ety.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eua.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fzk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fzl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fzo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fzp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gac.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gad.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gae.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gaf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gag.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gah.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gai.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gaj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gam.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gan.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gaq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gar.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.grh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.grj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gsn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gsp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.5.bhp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bhr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bjl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bjn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.180.13.brs.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bxm.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.efo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.efq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.egc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ege.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eig.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eii.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fgo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fgp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fgs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fgt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhi.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fhv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fud.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fuf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fvj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fvl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.5.sj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.sl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.uf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.uh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.21.bcm.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.efo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.efq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.egc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ege.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eig.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eii.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fgo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fgp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fgs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fgt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhi.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fhv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fud.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fuf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fvj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fvl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.5.sj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.sl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.uf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.uh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.3.efo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.efq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.egc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ege.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eig.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eii.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fgo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fgp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fgs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fgt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhi.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhn.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fhv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fud.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fuf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fvj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fvl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.5.sj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.sl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.uf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.uh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |