Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $12^{12}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.144.7.8 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&3\\2&11\end{bmatrix}$, $\begin{bmatrix}5&1\\0&7\end{bmatrix}$, $\begin{bmatrix}7&3\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_4:D_4$ |
| 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: | $32$ |
Jacobian
| Conductor: | $2^{26}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 36.2.a.a, 48.2.a.a$^{2}$, 144.2.a.a$^{3}$, 144.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} - x y + x z - y w + v^{2} $ |
| $=$ | $x^{2} + x y + x z - u v$ | |
| $=$ | $x^{2} + x z + y w - t v$ | |
| $=$ | $t^{2} - t u + u^{2} - v^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 6 x^{5} z - 5 x^{4} y^{2} + 12 x^{4} z^{2} - 16 x^{3} y^{2} z + 16 x^{3} z^{3} + x^{2} y^{4} + \cdots + z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 12.72.3.ds.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -t+u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x+w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+X^{2}Y^{2}-4X^{2}YZ-2Y^{3}Z+X^{2}Z^{2}+8Y^{2}Z^{2}-2YZ^{3} $ |
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 |
|---|---|---|---|---|---|---|
| 12.48.3.r.1 | $12$ | $3$ | $3$ | $3$ | $0$ | $1^{4}$ |
| 12.72.2.o.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{5}$ |
| 12.72.2.q.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{5}$ |
| 12.72.3.dh.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 12.72.3.ds.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 12.72.3.du.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 12.72.4.v.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 12.72.4.y.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.288.13.y.1 | $12$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 12.288.13.z.1 | $12$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 24.288.13.ro.1 | $24$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 24.288.13.rp.1 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 24.288.17.ebe.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.288.17.ebw.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.288.17.ejk.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.288.17.ejy.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.288.17.fqa.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.288.17.fqa.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.288.17.fqb.1 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 24.288.17.fqb.2 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 24.288.17.geo.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.288.17.gfc.1 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 24.288.17.gog.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.288.17.goy.1 | $24$ | $2$ | $2$ | $17$ | $5$ | $1^{10}$ |
| 36.432.25.fk.1 | $36$ | $3$ | $3$ | $25$ | $11$ | $1^{12}\cdot2^{3}$ |
| 36.432.31.ev.1 | $36$ | $3$ | $3$ | $31$ | $8$ | $1^{24}$ |
| 60.288.13.lj.1 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.288.13.lk.1 | $60$ | $2$ | $2$ | $13$ | $3$ | $1^{6}$ |
| 60.720.55.zt.1 | $60$ | $5$ | $5$ | $55$ | $23$ | $1^{48}$ |
| 60.864.61.ckt.1 | $60$ | $6$ | $6$ | $61$ | $11$ | $1^{54}$ |
| 60.1440.109.ehz.1 | $60$ | $10$ | $10$ | $109$ | $48$ | $1^{102}$ |
| 84.288.13.hq.1 | $84$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 84.288.13.hr.1 | $84$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.hkc.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.13.hkd.1 | $120$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 120.288.17.bvza.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bvzg.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bwcu.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.bwcw.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cdda.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cdda.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cddb.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cddb.2 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cily.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cima.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cipo.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 120.288.17.cipu.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 132.288.13.eq.1 | $132$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 132.288.13.er.1 | $132$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 156.288.13.fk.1 | $156$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 156.288.13.fl.1 | $156$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 168.288.13.dyu.1 | $168$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 168.288.13.dyv.1 | $168$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 168.288.17.bpco.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bpcu.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bpgi.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bpgk.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.buoc.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.buoc.2 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.buod.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.buod.2 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bxlc.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bxle.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bxos.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 168.288.17.bxoy.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 204.288.13.fg.1 | $204$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 204.288.13.fh.1 | $204$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 228.288.13.eu.1 | $228$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 228.288.13.ev.1 | $228$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 264.288.13.dpo.1 | $264$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 264.288.13.dpp.1 | $264$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 264.288.17.bpfx.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bpgd.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bpjr.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bpjt.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.burl.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.burl.2 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.burm.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.burm.2 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bxol.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bxon.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bxsb.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 264.288.17.bxsh.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 276.288.13.eq.1 | $276$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 276.288.13.er.1 | $276$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 312.288.13.dua.1 | $312$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 312.288.13.dub.1 | $312$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 312.288.17.bpga.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bpgg.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bpju.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bpjw.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.burm.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.burm.2 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.burn.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.burn.2 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bxom.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bxoo.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bxsc.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 312.288.17.bxsi.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |