Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| 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: | 12.96.1.110 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}7&11\\0&11\end{bmatrix}$, $\begin{bmatrix}11&2\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $D_6:C_4$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.j.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y z $ |
| $=$ | $15 x^{2} + 3 y^{2} + 15 y z + 27 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 10 x^{2} z^{2} - 3 y^{2} z^{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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^3}\cdot\frac{271724544yz^{11}+121927680yz^{9}w^{2}+19035648yz^{7}w^{4}+1154304yz^{5}w^{6}+18864yz^{3}w^{8}+72yzw^{10}+268738560z^{12}+110979072z^{10}w^{2}+13996800z^{8}w^{4}+387072z^{6}w^{6}-25344z^{4}w^{8}-504z^{2}w^{10}-w^{12}}{w^{2}z^{6}(5832yz^{3}+54yzw^{2}+5832z^{4}-189z^{2}w^{2}-w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.48.1.j.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+10X^{2}Z^{2}-3Y^{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.f.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.f.1.4 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.h.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.h.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.1-12.i.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.48.1-12.i.1.4 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.288.5-12.q.1.2 | $12$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.3-24.fk.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.fk.2.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.fl.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.fl.2.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.5-36.j.1.4 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 36.288.9-36.s.1.2 | $36$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 36.288.9-36.t.1.3 | $36$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 60.480.17-60.r.1.5 | $60$ | $5$ | $5$ | $17$ | $6$ | $1^{16}$ |
| 60.576.17-60.z.1.1 | $60$ | $6$ | $6$ | $17$ | $3$ | $1^{16}$ |
| 60.960.33-60.bz.1.12 | $60$ | $10$ | $10$ | $33$ | $13$ | $1^{32}$ |
| 120.192.3-120.lq.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lq.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lr.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lr.2.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.jg.1.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.jg.2.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.jh.1.13 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.jh.2.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.jg.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.jg.2.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.jh.1.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.jh.2.9 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.lq.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.lq.2.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.lr.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.lr.2.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |