Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ 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: | 12B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.2.4 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&8\\8&1\end{bmatrix}$, $\begin{bmatrix}7&4\\8&1\end{bmatrix}$, $\begin{bmatrix}11&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&10\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times \SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 12.36.2.b.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 36.2.a.a$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} z - y^{2} w + z^{2} w $ |
| $=$ | $x y w - 4 w^{3}$ | |
| $=$ | $x y z - 4 z w^{2}$ | |
| $=$ | $x y^{2} - 4 y w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - x^{2} y^{2} - 2 y z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} + 1 $ |
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.
| Embedded model |
|---|
| $(1:0:0:0)$, $(0:-1:1:0)$, $(0:0:1:0)$, $(0:1:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{256x^{4}w^{4}+256y^{2}w^{6}+z^{8}+48z^{5}w^{3}+512z^{2}w^{6}}{w^{6}z^{2}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-X^{2}Y^{2}-2YZ^{3} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.36.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{8}y^{2}z+w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 4.24.0-4.b.1.1 | $4$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 4.24.0-4.b.1.1 | $4$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.36.1-6.a.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 12.36.1-12.c.1.7 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.3-12.b.1.4 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.d.1.3 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.k.1.8 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.l.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.4-12.b.1.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.e.1.5 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.h.1.3 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.k.1.4 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.3-24.e.1.11 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.k.1.19 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.be.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.bh.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.4-24.c.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.h.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.m.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.n.1.21 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.q.1.22 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.r.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.s.1.37 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.s.2.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.t.1.54 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.t.2.10 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.u.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.u.2.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.v.1.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.v.2.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.w.1.16 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.w.2.16 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.x.1.13 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.x.2.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.y.1.42 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.y.2.2 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.z.1.43 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.z.2.9 | $24$ | $2$ | $2$ | $4$ | $0$ | $2$ |
| 24.144.4-24.be.1.17 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.bl.1.17 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.cc.1.31 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.cd.1.37 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.cg.1.31 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.5-24.c.1.44 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 24.144.5-24.d.1.32 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 24.144.5-24.g.1.34 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 24.144.5-24.h.1.32 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 24.144.5-24.k.1.17 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 24.144.5-24.l.1.23 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 24.144.5-24.o.1.17 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 24.144.5-24.p.1.24 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 36.216.8-36.b.1.6 | $36$ | $3$ | $3$ | $8$ | $1$ | $1^{6}$ |
| 36.648.22-36.f.1.3 | $36$ | $9$ | $9$ | $22$ | $5$ | $1^{18}\cdot2$ |
| 60.144.3-60.o.1.11 | $60$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 60.144.3-60.p.1.5 | $60$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 60.144.3-60.w.1.11 | $60$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 60.144.3-60.x.1.12 | $60$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 60.144.4-60.c.1.13 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.f.1.11 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.m.1.11 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.144.4-60.p.1.11 | $60$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 60.360.14-60.b.1.10 | $60$ | $5$ | $5$ | $14$ | $5$ | $1^{12}$ |
| 60.432.15-60.b.1.9 | $60$ | $6$ | $6$ | $15$ | $0$ | $1^{13}$ |
| 60.720.27-60.cr.1.1 | $60$ | $10$ | $10$ | $27$ | $9$ | $1^{25}$ |
| 84.144.3-84.k.1.11 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.l.1.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.s.1.12 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.t.1.12 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.4-84.c.1.13 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.f.1.7 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.m.1.11 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.p.1.11 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.3-120.bi.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bl.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cg.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cj.1.39 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.4-120.e.1.40 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.l.1.40 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bc.1.76 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bd.1.77 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bg.1.78 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bh.1.77 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bi.1.66 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bi.2.65 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bj.1.99 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bj.2.97 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bk.1.73 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bk.2.45 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bl.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bl.2.43 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bm.1.73 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bm.2.45 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bn.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bn.2.43 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bo.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bo.2.65 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bp.1.82 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bp.2.97 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.bu.1.43 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.cb.1.43 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.eo.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ep.1.78 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.es.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.et.1.67 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.5-120.c.1.78 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.d.1.70 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.g.1.70 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.h.1.70 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.k.1.50 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.l.1.53 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.o.1.50 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5-120.p.1.56 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.3-132.k.1.11 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.l.1.5 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.s.1.12 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.t.1.12 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.4-132.c.1.13 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.f.1.11 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.m.1.10 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.p.1.11 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.3-156.k.1.11 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.l.1.5 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.s.1.11 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.t.1.12 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.4-156.c.1.5 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.f.1.7 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.m.1.11 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.p.1.11 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.3-168.be.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bh.1.39 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cc.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cf.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.4-168.e.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.l.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bc.1.75 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bd.1.77 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bg.1.76 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bh.1.77 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bi.1.65 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bi.2.34 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bj.1.97 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bj.2.99 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bk.1.74 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bk.2.78 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bl.1.70 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bl.2.78 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bm.1.74 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bm.2.78 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bn.1.68 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bn.2.76 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bo.1.65 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bo.2.67 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bp.1.97 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bp.2.82 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.bu.1.44 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.cb.1.44 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.eo.1.39 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ep.1.78 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.es.1.39 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.et.1.67 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.5-168.c.1.78 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5-168.d.1.51 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5-168.g.1.69 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5-168.h.1.51 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5-168.k.1.67 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
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