Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | 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: | 12F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.1.28 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&3\\0&11\end{bmatrix}$, $\begin{bmatrix}11&5\\6&11\end{bmatrix}$, $\begin{bmatrix}11&6\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2:D_{12}$ |
| Contains $-I$: | no $\quad$ (see 12.24.1.i.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $96$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 219x + 1190 $ |
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.
| Weierstrass model |
|---|
| $(0:1:0)$, $(-17:0:1)$, $(7:0:1)$, $(10:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{12x^{2}y^{6}-24543x^{2}y^{4}z^{2}+13401936x^{2}y^{2}z^{4}-2423430009x^{2}z^{6}-294xy^{6}z+417960xy^{4}z^{3}-228836745xy^{2}z^{5}+41794764102xz^{7}-y^{8}+2496y^{6}z^{2}-2288088y^{4}z^{4}+1024189596y^{2}z^{6}-175992060609z^{8}}{z^{4}y^{2}(24x^{2}-408xz-y^{2}+1680z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.24.0-6.a.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.d.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.f.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.j.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.k.1.5 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.144.3-12.cf.1.2 | $12$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 24.96.1-24.cr.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.dx.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ij.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.im.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 36.144.3-36.u.1.4 | $36$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 36.144.5-36.h.1.5 | $36$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 36.144.5-36.l.1.5 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 60.96.1-60.bg.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.bh.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.bk.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.1-60.bl.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.240.9-60.dq.1.1 | $60$ | $5$ | $5$ | $9$ | $1$ | $1^{8}$ |
| 60.288.9-60.fu.1.17 | $60$ | $6$ | $6$ | $9$ | $2$ | $1^{8}$ |
| 60.480.17-60.ne.1.7 | $60$ | $10$ | $10$ | $17$ | $3$ | $1^{16}$ |
| 84.96.1-84.bg.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bh.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bk.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bl.1.1 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.384.13-84.bb.1.1 | $84$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 120.96.1-120.bym.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.byp.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.byy.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.bzb.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.96.1-132.bg.1.5 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.96.1-132.bh.1.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.96.1-132.bk.1.1 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.96.1-132.bl.1.5 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.96.1-156.bg.1.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.96.1-156.bh.1.5 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.96.1-156.bk.1.5 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.96.1-156.bl.1.1 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byk.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byn.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byw.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byz.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.96.1-204.bg.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.96.1-204.bh.1.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.96.1-204.bk.1.1 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.96.1-204.bl.1.5 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1-228.bg.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1-228.bh.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1-228.bk.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.96.1-228.bl.1.1 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byk.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byn.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byw.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.byz.1.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.96.1-276.bg.1.5 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.96.1-276.bh.1.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.96.1-276.bk.1.1 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.96.1-276.bl.1.5 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bym.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byp.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.byy.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bzb.1.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |