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: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.2.11 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&4\\4&5\end{bmatrix}$, $\begin{bmatrix}3&10\\8&9\end{bmatrix}$, $\begin{bmatrix}3&10\\10&11\end{bmatrix}$, $\begin{bmatrix}7&8\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times \SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 12.36.2.d.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $8$ |
| 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 $ | $=$ | $ 3 x^{2} y + x z^{2} + y w^{2} $ |
| $=$ | $3 x^{2} w + 12 x y z + w^{3}$ | |
| $=$ | $12 y^{2} w - z w^{2}$ | |
| $=$ | $12 y^{2} z - z^{2} w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{3} y + 27 y^{2} z^{2} + z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 27 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:1:0)$, $(1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{27x^{6}z^{2}+9x^{4}z^{2}w^{2}+6x^{2}z^{2}w^{4}+3xyw^{6}+16z^{8}-8z^{5}w^{3}+2z^{2}w^{6}}{w^{4}z^{2}(3x^{2}+w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{18}x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{3}Y+27Y^{2}Z^{2}+Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.36.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{24}xw^{2}+y^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-12.b.1.1 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.36.1-6.a.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 12.36.1-6.a.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
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.f.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.h.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.o.1.5 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.3-12.p.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 12.144.4-12.d.1.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.d.1.3 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.e.1.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.e.1.7 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.n.1.3 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.n.1.4 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.o.1.2 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 12.144.4-12.o.1.3 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.3-24.q.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.w.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.bq.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.bt.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.4-24.g.1.4 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.g.1.15 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.j.1.4 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.j.1.15 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bu.1.5 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bu.1.10 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.bx.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.bx.1.10 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 36.216.8-36.d.1.3 | $36$ | $3$ | $3$ | $8$ | $1$ | $1^{6}$ |
| 36.648.22-36.h.1.2 | $36$ | $9$ | $9$ | $22$ | $6$ | $1^{18}\cdot2$ |
| 60.144.3-60.s.1.6 | $60$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 60.144.3-60.t.1.7 | $60$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 60.144.3-60.be.1.7 | $60$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 60.144.3-60.bf.1.7 | $60$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 60.144.4-60.i.1.5 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.i.1.11 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.j.1.2 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.j.1.13 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.s.1.2 | $60$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 60.144.4-60.s.1.7 | $60$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 60.144.4-60.t.1.3 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.144.4-60.t.1.6 | $60$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 60.360.14-60.d.1.5 | $60$ | $5$ | $5$ | $14$ | $5$ | $1^{12}$ |
| 60.432.15-60.d.1.9 | $60$ | $6$ | $6$ | $15$ | $1$ | $1^{13}$ |
| 60.720.27-60.ct.1.3 | $60$ | $10$ | $10$ | $27$ | $10$ | $1^{25}$ |
| 84.144.3-84.o.1.6 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.p.1.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.ba.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.3-84.bb.1.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.144.4-84.i.1.5 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.i.1.11 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.j.1.2 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.j.1.13 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.s.1.3 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.s.1.10 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.t.1.3 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 84.144.4-84.t.1.10 | $84$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.3-120.bu.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bx.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.de.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.dh.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.4-120.u.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.u.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.x.1.4 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.x.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ck.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ck.1.27 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.cn.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.cn.1.27 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.3-132.o.1.6 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.p.1.5 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.ba.1.8 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.3-132.bb.1.5 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.144.4-132.i.1.7 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.i.1.9 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.j.1.2 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.j.1.13 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.s.1.3 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.s.1.10 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.t.1.3 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 132.144.4-132.t.1.10 | $132$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.3-156.o.1.6 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.p.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.ba.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.3-156.bb.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.144.4-156.i.1.5 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.i.1.11 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.j.1.2 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.j.1.13 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.s.1.2 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.s.1.11 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.t.1.3 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 156.144.4-156.t.1.10 | $156$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.3-168.bq.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bt.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.da.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.dd.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.4-168.u.1.3 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.u.1.29 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.x.1.3 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.x.1.29 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ck.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ck.1.28 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.cn.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.cn.1.28 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.3-204.o.1.7 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.144.3-204.p.1.8 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.144.3-204.ba.1.7 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.144.3-204.bb.1.7 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.144.4-204.i.1.1 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.i.1.14 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.j.1.3 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.j.1.9 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.s.1.11 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.s.1.13 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.t.1.10 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 204.144.4-204.t.1.13 | $204$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.3-228.o.1.5 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.144.3-228.p.1.7 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.144.3-228.ba.1.8 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.144.3-228.bb.1.7 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.144.4-228.i.1.5 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.i.1.11 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.j.1.2 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.j.1.13 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.s.1.2 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.s.1.11 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.t.1.3 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.144.4-228.t.1.10 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.3-264.bq.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.bt.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.da.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.dd.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.4-264.u.1.3 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.u.1.29 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.x.1.3 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.x.1.29 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ck.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ck.1.28 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.cn.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.cn.1.28 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.3-276.o.1.5 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.144.3-276.p.1.7 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.144.3-276.ba.1.8 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.144.3-276.bb.1.7 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.144.4-276.i.1.7 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.i.1.9 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.j.1.2 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.j.1.13 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.s.1.2 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.s.1.11 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.t.1.3 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 276.144.4-276.t.1.10 | $276$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.3-312.bq.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.bt.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.da.1.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.dd.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.4-312.u.1.4 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.u.1.30 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.x.1.4 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.x.1.30 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ck.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ck.1.27 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.cn.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.cn.1.27 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |