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: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.1.68 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&0\\2&7\end{bmatrix}$, $\begin{bmatrix}5&6\\0&11\end{bmatrix}$, $\begin{bmatrix}7&6\\2&5\end{bmatrix}$, $\begin{bmatrix}11&6\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.d.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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 39x - 70 $ |
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 |
|---|
| $(-5:0:1)$, $(-2:0:1)$, $(7:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2}\cdot\frac{48x^{2}y^{14}+20603970x^{2}y^{12}z^{2}+596598186024x^{2}y^{10}z^{4}+2591415006608565x^{2}y^{8}z^{6}+2999663765443954440x^{2}y^{6}z^{8}+1342582209064924248249x^{2}y^{4}z^{10}+253565669955443818968540x^{2}y^{2}z^{12}+16997644367594655055874037x^{2}z^{14}+7572xy^{14}z+669231720xy^{12}z^{3}+12144823339167xy^{10}z^{5}+33418801143993474xy^{8}z^{7}+30097343519700840576xy^{6}z^{9}+11462543023009458260880xy^{4}z^{11}+1932346127304845969021721xy^{2}z^{13}+118983545466698516454699030xz^{15}+y^{16}+300720y^{14}z^{2}+24011388900y^{12}z^{4}+183330009767520y^{10}z^{6}+297992456081381412y^{8}z^{8}+173882791072168531488y^{6}z^{10}+43785663587859745263402y^{4}z^{12}+4739058019096060967028648y^{2}z^{14}+169976622882007544198261121z^{16}}{zy^{4}(693x^{2}y^{8}z-31104x^{2}y^{6}z^{3}-8398080x^{2}y^{4}z^{5}-362797056x^{2}y^{2}z^{7}+78364164096x^{2}z^{9}+xy^{10}+4662xy^{8}z^{2}+342144xy^{6}z^{4}+57106944xy^{4}z^{6}+3990767616xy^{2}z^{8}-391820820480xz^{10}+44y^{10}z-6975y^{8}z^{3}-311040y^{6}z^{5}+26873856y^{4}z^{7}-16688664576y^{2}z^{9}-1097098297344z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.a.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-6.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.192.1-12.d.1.6 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.192.1-12.d.1.8 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.192.1-12.d.2.7 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.192.1-12.d.2.8 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.192.3-12.d.1.3 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.192.3-12.d.1.6 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.192.3-12.e.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.192.3-12.e.1.4 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.192.3-12.i.1.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.192.3-12.i.1.3 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.192.3-12.i.2.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.192.3-12.i.2.6 | $12$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 12.288.5-12.e.1.5 | $12$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.1-24.co.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.co.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.co.2.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.co.2.13 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.be.1.6 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.be.1.13 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.bh.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.bh.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cb.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.cb.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.cb.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.cb.2.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 36.288.5-36.d.1.10 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 36.288.9-36.d.1.3 | $36$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 36.288.9-36.h.1.2 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{8}$ |
| 60.192.1-60.d.1.14 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.d.1.15 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.d.2.13 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.d.2.16 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.3-60.i.1.12 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.192.3-60.i.1.15 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.192.3-60.j.1.8 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.j.1.11 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.192.3-60.n.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.n.1.10 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.n.2.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.192.3-60.n.2.9 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.480.17-60.d.1.1 | $60$ | $5$ | $5$ | $17$ | $2$ | $1^{16}$ |
| 60.576.17-60.d.1.9 | $60$ | $6$ | $6$ | $17$ | $2$ | $1^{16}$ |
| 60.960.33-60.p.1.15 | $60$ | $10$ | $10$ | $33$ | $6$ | $1^{32}$ |
| 84.192.1-84.d.1.13 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.d.1.16 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.d.2.13 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.1-84.d.2.15 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.192.3-84.i.1.10 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.i.1.16 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.j.1.2 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.j.1.14 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.n.1.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.n.1.15 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.n.2.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.n.2.15 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.1-120.lk.1.25 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lk.1.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lk.2.27 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lk.2.30 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ei.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ei.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.el.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.el.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fh.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fh.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fh.2.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fh.2.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.1-132.d.1.14 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.192.1-132.d.1.15 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.192.1-132.d.2.13 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.192.1-132.d.2.16 | $132$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 132.192.3-132.i.1.8 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.i.1.14 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.j.1.6 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.j.1.12 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.n.1.7 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.n.1.11 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.n.2.5 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.n.2.9 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.1-156.d.1.12 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.192.1-156.d.1.15 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.192.1-156.d.2.13 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.192.1-156.d.2.15 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 156.192.3-156.i.1.10 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.i.1.14 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.j.1.2 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.j.1.14 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.n.1.5 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.n.1.15 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.n.2.3 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.n.2.15 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.1-168.lk.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lk.1.23 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lk.2.23 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lk.2.27 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.dk.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.dk.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.dn.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.dn.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ej.1.5 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ej.1.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ej.2.3 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ej.2.9 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.1-204.d.1.14 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.192.1-204.d.1.15 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.192.1-204.d.2.13 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.192.1-204.d.2.16 | $204$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 204.192.3-204.i.1.12 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.i.1.15 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.j.1.7 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.j.1.12 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.n.1.4 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.n.1.7 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.n.2.3 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.n.2.5 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.1-228.d.1.13 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.d.1.15 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.d.2.13 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.1-228.d.2.15 | $228$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 228.192.3-228.i.1.10 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.i.1.14 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.j.1.2 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.j.1.14 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.n.1.5 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.n.1.15 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.n.2.5 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.n.2.15 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.lk.1.23 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lk.1.29 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lk.2.17 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.lk.2.26 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.dk.1.8 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.dk.1.23 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.dn.1.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.dn.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ej.1.6 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ej.1.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ej.2.6 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ej.2.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.1-276.d.1.13 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.192.1-276.d.1.16 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.192.1-276.d.2.13 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.192.1-276.d.2.16 | $276$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 276.192.3-276.i.1.8 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.i.1.14 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.j.1.6 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.j.1.12 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.n.1.7 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.n.1.13 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.n.2.5 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.n.2.9 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.lk.1.26 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lk.1.29 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lk.2.26 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.lk.2.27 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ei.1.24 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ei.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.el.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.el.1.28 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fh.1.9 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fh.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fh.2.5 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.fh.2.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |