Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 12.36.1.17 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&0\\0&11\end{bmatrix}$, $\begin{bmatrix}3&7\\10&3\end{bmatrix}$, $\begin{bmatrix}5&3\\0&1\end{bmatrix}$, $\begin{bmatrix}5&3\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2.D_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $128$ |
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 |
|---|
| $(0:1:0)$, $(5:0:1)$, $(2:0:1)$, $(-7:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^6}\cdot\frac{18x^{2}y^{10}+1782x^{2}y^{8}z^{2}-3201768x^{2}y^{6}z^{4}-1250303526x^{2}y^{4}z^{6}+1655388759546x^{2}y^{2}z^{8}-288881335977759x^{2}z^{10}+63xy^{10}z-2268xy^{8}z^{3}+24439725xy^{6}z^{5}+20927437992xy^{4}z^{7}-14096527263423xy^{2}z^{9}+2025240175780290xz^{11}+y^{12}+18y^{10}z^{2}+552906y^{8}z^{4}+29668842y^{6}z^{6}-193010690997y^{4}z^{8}+53218300855302y^{2}z^{10}-2902580605134531z^{12}}{z^{4}y^{6}(6x^{2}-51xz-y^{2}+78z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.18.0.j.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.18.1.g.1 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.72.1.o.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.72.1.r.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.72.3.m.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.bc.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.bt.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.by.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.ce.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.ch.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.1.bw.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cf.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.dm.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.hh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ls.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.nb.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qx.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 36.108.5.b.1 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 36.108.5.n.1 | $36$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 60.72.1.eh.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.ei.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.3.ss.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.su.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.ta.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.tc.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.tg.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.3.th.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.180.13.mk.1 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{12}$ |
| 60.216.13.ox.1 | $60$ | $6$ | $6$ | $13$ | $2$ | $1^{12}$ |
| 60.360.25.cbi.1 | $60$ | $10$ | $10$ | $25$ | $7$ | $1^{24}$ |
| 84.72.1.bu.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.72.1.bv.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.72.3.oy.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.pa.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.pg.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.pi.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.pm.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.pn.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.21.ja.1 | $84$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 120.72.1.og.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.emn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.enb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eor.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.1.bs.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.72.1.bt.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.72.3.oy.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.pa.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.pg.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.pi.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.pm.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.pn.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.1.bu.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.72.1.bv.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.72.3.oy.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.pa.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.pg.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.pi.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.pm.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.pn.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.1.ga.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.gd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.eav.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edn.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eee.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eeh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.1.bu.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.1.bv.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.72.3.oy.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.pa.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.pg.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.pi.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.pm.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.pn.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.1.bu.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.72.1.bv.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.72.3.oy.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.pa.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.pg.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.pi.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.pm.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.pn.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.1.fw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.eav.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edn.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eee.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eeh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.1.bs.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.72.1.bt.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.72.3.oy.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.pa.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.pg.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.pi.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.pm.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.pn.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.ga.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.gd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.eav.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edn.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eee.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eeh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |