Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.1.114 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&9\\6&5\end{bmatrix}$, $\begin{bmatrix}7&7\\6&11\end{bmatrix}$, $\begin{bmatrix}11&6\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_4\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.e.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| Full 12-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $5 x^{2} + y^{2} + 5 y z + 9 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 10 x^{2} z^{2} + y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{372736yz^{11}-501760yz^{9}w^{2}+235008yz^{7}w^{4}-42752yz^{5}w^{6}+2096yz^{3}w^{8}-24yzw^{10}+368640z^{12}-456704z^{10}w^{2}+172800z^{8}w^{4}-14336z^{6}w^{6}-2816z^{4}w^{8}+168z^{2}w^{10}-w^{12}}{w^{2}z^{6}(648yz^{3}-18yzw^{2}+648z^{4}+63z^{2}w^{2}-w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.48.1.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+10X^{2}Z^{2}+Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.d.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.d.1.10 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.f.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.1-12.k.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.48.1-12.k.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.192.3-12.j.1.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.192.3-12.j.2.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.288.5-12.i.1.2 | $12$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.3-24.di.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.di.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dj.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dj.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dk.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.dk.2.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.5-24.u.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.v.1.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.w.1.6 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.x.1.4 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 36.288.5-36.e.1.4 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 36.288.9-36.i.1.4 | $36$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 36.288.9-36.j.1.5 | $36$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 60.192.3-60.w.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.w.2.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.480.17-60.i.1.8 | $60$ | $5$ | $5$ | $17$ | $3$ | $1^{16}$ |
| 60.576.17-60.m.1.1 | $60$ | $6$ | $6$ | $17$ | $0$ | $1^{16}$ |
| 60.960.33-60.bc.1.20 | $60$ | $10$ | $10$ | $33$ | $6$ | $1^{32}$ |
| 84.192.3-84.w.1.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.w.2.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ig.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ig.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ih.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ih.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ii.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ii.2.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.u.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.v.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.w.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.x.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.192.3-132.w.1.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.w.2.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.w.1.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.w.2.2 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gk.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gk.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gl.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gl.2.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gm.1.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.gm.2.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.5-168.bg.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.bh.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.bi.1.13 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.bj.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.192.3-204.w.1.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.w.2.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.w.1.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.w.2.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gk.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gk.2.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gl.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gl.2.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gm.1.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.gm.2.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.5-264.u.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.v.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.w.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.x.1.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.192.3-276.w.1.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.w.2.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ig.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ig.2.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ih.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ih.2.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ii.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ii.2.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.u.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.v.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.w.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.x.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |