Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12C4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.4.20 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&2\\10&7\end{bmatrix}$, $\begin{bmatrix}3&11\\2&9\end{bmatrix}$, $\begin{bmatrix}11&2\\8&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $Q_8:D_4$ |
| 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: | $64$ |
Jacobian
| Conductor: | $2^{13}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}$ |
| Newforms: | 24.2.a.a, 36.2.a.a, 144.2.a.a, 144.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 2 x^{2} + 16 y^{2} - 3 z^{2} - w^{2} $ |
| $=$ | $6 x^{2} y - 3 x z^{2} + x w^{2} + 3 y z^{2} + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{6} - 4 x^{5} y + 4 x^{4} y^{2} + 9 x^{4} z^{2} + 26 x^{3} y^{3} - 36 x^{3} y z^{2} + \cdots + 36 y^{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 canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{27216xyz^{10}-190512xyz^{8}w^{2}+1060128xyz^{6}w^{4}-353376xyz^{4}w^{6}+7056xyz^{2}w^{8}-112xyw^{10}+54432y^{2}z^{10}-552096y^{2}z^{8}w^{2}+3053376y^{2}z^{6}w^{4}+1017792y^{2}z^{4}w^{6}-20448y^{2}z^{2}w^{8}+224y^{2}w^{10}-13851z^{12}+109350z^{10}w^{2}-733293z^{8}w^{4}-529740z^{6}w^{6}-81477z^{4}w^{8}+1350z^{2}w^{10}-19w^{12}}{27216xyz^{10}+89424xyz^{8}w^{2}+33696xyz^{6}w^{4}-11232xyz^{4}w^{6}-3312xyz^{2}w^{8}-112xyw^{10}+54432y^{2}z^{10}+7776y^{2}z^{8}w^{2}-119232y^{2}z^{6}w^{4}-39744y^{2}z^{4}w^{6}+288y^{2}z^{2}w^{8}+224y^{2}w^{10}-13851z^{12}-21870z^{10}w^{2}+1539z^{8}w^{4}+4860z^{6}w^{6}+171z^{4}w^{8}-270z^{2}w^{10}-19w^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.1.bw.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
| 12.36.1.bz.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
| 12.36.2.bx.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.7.bu.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.bx.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.co.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 12.144.7.cq.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 24.144.7.bgj.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 24.144.7.bgz.1 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.144.7.bny.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 24.144.7.bor.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 24.144.9.bqn.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.144.9.bqx.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.brf.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.brl.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.crp.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.144.9.crv.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.cvj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.144.9.cvl.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.dlz.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.144.9.dmb.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 24.144.9.dpr.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 24.144.9.dpx.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{5}$ |
| 24.144.9.egj.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.egp.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.egx.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 24.144.9.ehh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 36.216.16.bf.1 | $36$ | $3$ | $3$ | $16$ | $4$ | $1^{12}$ |
| 36.648.46.bt.1 | $36$ | $9$ | $9$ | $46$ | $17$ | $1^{24}\cdot2^{9}$ |
| 60.144.7.jl.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.144.7.jp.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.le.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
| 60.144.7.li.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
| 60.360.28.dk.1 | $60$ | $5$ | $5$ | $28$ | $11$ | $1^{24}$ |
| 60.432.31.hm.1 | $60$ | $6$ | $6$ | $31$ | $4$ | $1^{27}$ |
| 60.720.55.ym.1 | $60$ | $10$ | $10$ | $55$ | $22$ | $1^{51}$ |
| 84.144.7.gz.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.hd.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.ib.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 84.144.7.if.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gns.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.gpc.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.hdq.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.7.hfa.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.144.9.llp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.llv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lmh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.lmj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.meb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.meh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.mhf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.mhh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.omd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.omf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.opd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.opj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.pqx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.pqz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.prl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.prr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.144.7.gn.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.gr.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.hp.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 132.144.7.ht.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.il.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.ip.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.jp.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.144.7.jt.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.fqb.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.frl.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.gcj.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.7.gdt.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.144.9.kbp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kbv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kch.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kcj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kub.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kuh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kxf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.kxh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mpn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mpp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.msn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mst.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.nuh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.nuj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.nuv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.nvb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.144.7.hz.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.id.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.jd.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 204.144.7.jh.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.gz.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.hd.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.ib.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 228.144.7.if.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fpp.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.fqz.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.gbx.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.7.gdh.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.9.ker.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kex.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kfj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kfl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kxd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.kxj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.lah.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.laj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mub.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mud.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mxb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mxh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.nyv.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.nyx.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.nzj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.nzp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.144.7.gn.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.gr.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.hp.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 276.144.7.ht.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.fzj.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.gat.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.glx.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.7.gnh.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.144.9.kbx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kcd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kcp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kcr.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kuj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kup.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kxn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.kxp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mpv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mpx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.msv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mtb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.nup.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.nur.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.nvd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.nvj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |