Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12V1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.1.89 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&0\\6&7\end{bmatrix}$, $\begin{bmatrix}5&6\\6&7\end{bmatrix}$, $\begin{bmatrix}7&8\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.96.1.a.2 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $24$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{8x^{2}y^{30}+176328x^{2}y^{28}z^{2}+57062760x^{2}y^{26}z^{4}+751512584x^{2}y^{24}z^{6}+2155992936x^{2}y^{22}z^{8}+2690410600x^{2}y^{20}z^{10}+8578240968x^{2}y^{18}z^{12}+39436660008x^{2}y^{16}z^{14}+88915974552x^{2}y^{14}z^{16}+95486976856x^{2}y^{12}z^{18}+27036105912x^{2}y^{10}z^{20}-46490190312x^{2}y^{8}z^{22}-55788536584x^{2}y^{6}z^{24}-25957171848x^{2}y^{4}z^{26}-5423886824x^{2}y^{2}z^{28}-387420488x^{2}z^{30}+728xy^{30}z+1013688xy^{28}z^{3}+234900552xy^{26}z^{5}+1623139880xy^{24}z^{7}+4389622488xy^{22}z^{9}+4808593592xy^{20}z^{11}+1613615112xy^{18}z^{13}+22675872744xy^{16}z^{15}+112244580936xy^{14}z^{17}+231775406056xy^{12}z^{19}+256087963800xy^{10}z^{21}+155528084280xy^{8}z^{23}+44165950472xy^{6}z^{25}+936xy^{4}z^{27}-2711943400xy^{2}z^{29}-387420488xz^{31}+y^{32}+4944y^{30}z^{2}+15695592y^{28}z^{4}+532853056y^{26}z^{6}+2153021884y^{24}z^{8}+3849513136y^{22}z^{10}+2729488536y^{20}z^{12}-13319673888y^{18}z^{14}-64498712058y^{16}z^{16}-133215120208y^{14}z^{18}-150890568360y^{12}z^{20}-93537492096y^{10}z^{22}-25182049604y^{8}z^{24}+2324537680y^{6}z^{26}+2711944360y^{4}z^{28}+387420512y^{2}z^{30}+z^{32}}{zy^{4}(y^{2}+z^{2})^{2}(28x^{2}y^{20}z-3056x^{2}y^{18}z^{3}-62817x^{2}y^{16}z^{5}+25856x^{2}y^{14}z^{7}+1414956x^{2}y^{12}z^{9}+2123520x^{2}y^{10}z^{11}+532186x^{2}y^{8}z^{13}+128x^{2}y^{6}z^{15}-64x^{2}y^{4}z^{17}-16x^{2}y^{2}z^{19}-x^{2}z^{21}-xy^{22}+233xy^{20}z^{2}+4615xy^{18}z^{4}-136361xy^{16}z^{6}-1213342xy^{14}z^{8}-1599354xy^{12}z^{10}+529754xy^{10}z^{12}+532362xy^{8}z^{14}+79xy^{6}z^{16}-79xy^{4}z^{18}-17xy^{2}z^{20}-xz^{22}-7y^{22}z-400y^{20}z^{3}+24871y^{18}z^{5}+320563y^{16}z^{7}+554982y^{14}z^{9}-536136y^{12}z^{11}-532918y^{10}z^{13}+886y^{8}z^{15}+65y^{6}z^{17}-80y^{4}z^{19}-17y^{2}z^{21}-z^{23})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.4 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.14 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.a.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.a.1.7 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.384.5-12.a.2.2 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.b.2.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.c.2.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.d.2.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.576.9-12.a.1.7 | $12$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.5-24.b.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.e.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bl.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bq.2.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.cs.1.11 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ct.2.14 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.da.1.11 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.db.1.13 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.u.1.14 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot4$ |
| 24.384.9-24.w.2.13 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot4$ |
| 24.384.9-24.dg.2.13 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot4$ |
| 24.384.9-24.di.2.13 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot4$ |
| 36.576.9-36.a.1.7 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 36.576.17-36.a.1.4 | $36$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot4^{2}$ |
| 36.576.17-36.e.1.6 | $36$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot4^{2}$ |
| 60.384.5-60.f.2.5 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.384.5-60.g.1.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.384.5-60.i.2.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.384.5-60.j.2.6 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.960.33-60.a.1.14 | $60$ | $5$ | $5$ | $33$ | $3$ | $1^{16}\cdot8^{2}$ |
| 60.1152.33-60.a.1.30 | $60$ | $6$ | $6$ | $33$ | $1$ | $1^{16}\cdot8^{2}$ |
| 60.1920.65-60.a.1.24 | $60$ | $10$ | $10$ | $65$ | $7$ | $1^{32}\cdot8^{4}$ |
| 84.384.5-84.f.2.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.g.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.i.2.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.j.2.7 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.in.2.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.is.2.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.je.2.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jj.2.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.lo.2.24 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.lq.2.24 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.me.2.28 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.mg.2.30 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.9-120.hb.1.16 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.hd.1.24 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.id.1.16 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.if.1.24 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.384.5-132.f.2.5 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.g.1.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.i.2.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.j.2.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.f.2.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.g.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.i.2.8 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.j.2.7 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.in.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.is.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.je.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jj.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.lo.1.21 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.lq.1.20 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.me.1.22 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.mg.1.21 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.9-168.hb.2.22 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.hd.2.21 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.id.2.21 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.if.2.22 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.384.5-204.f.2.3 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.g.1.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.i.2.4 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.j.2.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.f.2.6 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.g.1.2 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.i.2.8 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.j.2.7 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.in.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.is.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.je.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jj.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.lo.2.17 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.lq.2.26 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.me.2.17 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.mg.2.21 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.9-264.hb.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.hd.2.21 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.id.2.9 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.if.2.21 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.384.5-276.f.2.3 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.g.1.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.i.2.4 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.j.2.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.in.2.18 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.is.2.18 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.je.2.18 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jj.2.18 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.lo.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.lq.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.me.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.mg.2.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.9-312.hb.2.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.hd.2.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.id.1.8 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.if.2.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |