Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $12^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $4$ 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: | 12G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.1.2 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}4&7\\11&1\end{bmatrix}$, $\begin{bmatrix}5&8\\7&11\end{bmatrix}$, $\begin{bmatrix}11&10\\11&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_{24}:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y - z^{2} $ |
| $=$ | $27 x^{2} + 9 x z + y^{2} + 2 y z + y w + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + x^{3} y + 4 x^{3} z + x^{2} y^{2} + 2 x^{2} y z + 4 x^{2} z^{2} + 36 x z^{3} + 108 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -3^5\,\frac{3021xz^{5}-20100xz^{4}w+6144xz^{3}w^{2}+4374xz^{2}w^{3}+463y^{2}z^{4}-2268y^{2}z^{3}w+1620y^{2}zw^{3}+324y^{2}w^{4}+110yz^{5}-671yz^{4}w-810yz^{3}w^{2}-1620yz^{2}w^{3}+324yzw^{4}+324yw^{5}+208z^{6}-1163z^{5}w-2048z^{4}w^{2}+1458z^{3}w^{3}}{270xz^{5}-54xz^{4}w-432xz^{3}w^{2}+108xz^{2}w^{3}+108xzw^{4}+49y^{2}z^{4}+8y^{2}z^{3}w-24y^{2}z^{2}w^{2}+8y^{2}zw^{3}+4y^{2}w^{4}+26yz^{5}+74yz^{4}w-4yz^{3}w^{2}-8yz^{2}w^{3}+16yzw^{4}+4yw^{5}-41z^{6}-24z^{5}w+87z^{4}w^{2}+28z^{3}w^{3}-12z^{2}w^{4}+12zw^{5}+4w^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 12.12.0.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(12)$ | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.e.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.k.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.m.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.dk.1 | $12$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.h.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.n.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bf.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.5.iu.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
| 36.72.3.z.1 | $36$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 36.72.3.ba.1 | $36$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 36.216.15.ea.1 | $36$ | $9$ | $9$ | $15$ | $5$ | $1^{4}\cdot2^{5}$ |
| 60.48.3.bg.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bi.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bk.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bm.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.120.9.is.1 | $60$ | $5$ | $5$ | $9$ | $4$ | $1^{8}$ |
| 60.144.9.lk.1 | $60$ | $6$ | $6$ | $9$ | $2$ | $1^{8}$ |
| 60.240.17.ya.1 | $60$ | $10$ | $10$ | $17$ | $7$ | $1^{16}$ |
| 84.48.3.s.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.u.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.w.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.y.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.15.bs.1 | $84$ | $8$ | $8$ | $15$ | $?$ | not computed |
| 120.48.3.dj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.dp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.dv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.eb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.s.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.u.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.w.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.y.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.288.23.bu.1 | $132$ | $12$ | $12$ | $23$ | $?$ | not computed |
| 156.48.3.s.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.u.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.w.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.y.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.cp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.cv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.db.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.dh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.s.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.u.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.w.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.y.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.s.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.u.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.w.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.y.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.72.3.cu.1 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 252.72.3.cv.1 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 252.72.3.cw.1 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 264.48.3.cp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.cv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.db.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.dh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.s.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.u.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.w.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.y.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.cp.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.cv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.db.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.dh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |