Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $18$ | $\PSL_2$-index: | $18$ | ||||
| Genus: | $1 = 1 + \frac{ 18 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $2$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 12C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.18.1.5 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&7\\10&3\end{bmatrix}$, $\begin{bmatrix}7&9\\6&5\end{bmatrix}$, $\begin{bmatrix}7&10\\2&1\end{bmatrix}$, $\begin{bmatrix}11&6\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2.D_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} + x $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(0:0:1)$ | ||
| 256.a1 | $-8$ | $8000$ | $= 2^{6} \cdot 5^{3}$ | $8.987$ | $(1:-1:1)$, $(1:1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 18 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{192x^{2}y^{4}-160x^{2}y^{2}z^{2}+28x^{2}z^{4}+80xy^{2}z^{3}-28xz^{5}+64y^{6}-80y^{4}z^{2}+28y^{2}z^{4}+z^{6}}{z^{4}(x^{2}-xz+y^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 6.9.0.a.1 | $6$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.36.1.bc.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.1.bf.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.1.bo.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.1.br.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.36.2.e.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.m.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.u.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.x.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.bm.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.bp.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.bq.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.36.2.bt.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.1.dt.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.ec.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.fd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.1.fm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.2.p.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.bl.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.cz.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.di.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.ex.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.36.2.fg.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.fj.1 | $24$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 24.36.2.fs.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 36.54.3.i.1 | $36$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 36.162.11.d.1 | $36$ | $9$ | $9$ | $11$ | $4$ | $1^{4}\cdot2^{3}$ |
| 60.36.1.ep.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.er.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.ex.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1.ez.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.2.dd.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.df.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dh.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dj.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dn.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dp.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dr.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.36.2.dt.1 | $60$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 60.90.7.t.1 | $60$ | $5$ | $5$ | $7$ | $3$ | $1^{6}$ |
| 60.108.7.p.1 | $60$ | $6$ | $6$ | $7$ | $0$ | $1^{6}$ |
| 60.180.13.kt.1 | $60$ | $10$ | $10$ | $13$ | $5$ | $1^{12}$ |
| 84.36.1.ef.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.36.1.eh.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.36.1.en.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.36.1.ep.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.36.2.dd.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.df.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dh.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dj.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dn.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dp.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dr.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.36.2.dt.1 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.144.11.d.1 | $84$ | $8$ | $8$ | $11$ | $?$ | not computed |
| 120.36.1.oh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.on.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.pf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1.pl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.2.ka.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.kg.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.km.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ks.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.la.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.lg.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.lm.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.36.2.ls.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.1.eb.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.36.1.ed.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.36.1.ej.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.36.1.el.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.36.2.dd.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.df.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dh.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dj.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dn.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dp.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dr.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.36.2.dt.1 | $132$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 132.216.17.d.1 | $132$ | $12$ | $12$ | $17$ | $?$ | not computed |
| 156.36.1.ef.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.36.1.eh.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.36.1.en.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.36.1.ep.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.36.2.dd.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.df.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dh.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dj.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dn.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dp.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dr.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.36.2.dt.1 | $156$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 156.252.19.x.1 | $156$ | $14$ | $14$ | $19$ | $?$ | not computed |
| 168.36.1.nv.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.ob.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.ot.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.1.oz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.36.2.jz.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.kf.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.kl.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.kr.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.kz.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.lf.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.ll.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.36.2.lr.1 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.1.ec.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.36.1.ee.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.36.1.ek.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.36.1.em.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.36.2.dd.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.df.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dh.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dj.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dn.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dp.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dr.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 204.36.2.dt.1 | $204$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.1.ef.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.36.1.eh.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.36.1.en.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.36.1.ep.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.36.2.dd.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.df.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dh.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dj.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dn.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dp.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dr.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.36.2.dt.1 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.1.nm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.ns.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.ok.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.1.oq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.36.2.ka.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.kg.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.km.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ks.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.la.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.lg.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.lm.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 264.36.2.ls.1 | $264$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.1.eb.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.36.1.ed.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.36.1.ej.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.36.1.el.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.36.2.dd.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.df.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dh.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dj.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dn.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dp.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dr.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 276.36.2.dt.1 | $276$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.1.nv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.ob.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.ot.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.1.oz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.36.2.ka.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.kg.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.km.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ks.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.la.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.lg.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.lm.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.36.2.ls.1 | $312$ | $2$ | $2$ | $2$ | $?$ | not computed |