Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| 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 $4$ are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{2}$ | ||
| 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: | 12V1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.1.11 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&10\\0&7\end{bmatrix}$, $\begin{bmatrix}11&4\\6&1\end{bmatrix}$, $\begin{bmatrix}11&8\\6&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.96.1.d.2 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $2$ |
| Full 12-torsion field degree: | $24$ |
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} + 6x - 7 $ |
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 |
|---|
| $(1:0:1)$, $(0:1:0)$, $(4:9:1)$, $(4:-9:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^6}\cdot\frac{24x^{2}y^{30}-5832x^{2}y^{28}z^{2}-262440x^{2}y^{26}z^{4}+3761657496x^{2}y^{24}z^{6}+154547294328x^{2}y^{22}z^{8}+196263775989720x^{2}y^{20}z^{10}-46241818408887624x^{2}y^{18}z^{12}-2863069551014403528x^{2}y^{16}z^{14}-34083919561916814264x^{2}y^{14}z^{16}+256856416354490395176x^{2}y^{12}z^{18}+2786245628442160571784x^{2}y^{10}z^{20}+3436967022120883872072x^{2}y^{8}z^{22}+110686988610501407783208x^{2}y^{6}z^{24}+117588940319998615363272x^{2}y^{4}z^{26}-3342968955945965598553368x^{2}y^{2}z^{28}-6452218951107020797836312x^{2}z^{30}+24xy^{30}z+169128xy^{28}z^{3}-47816568xy^{26}z^{5}+13642208568xy^{24}z^{7}-5719550857704xy^{22}z^{9}-1735245350270424xy^{20}z^{11}+150794917589311560xy^{18}z^{13}+6360907393296876024xy^{16}z^{15}+34669443439204056072xy^{14}z^{17}+460392153897976610616xy^{12}z^{19}+9050728744651313218392xy^{10}z^{21}-38459849096142262767384xy^{8}z^{23}-580387136523518577075768xy^{6}z^{25}-227299713422528340863496xy^{4}z^{27}+1683301728799186134071256xy^{2}z^{29}-6452218951107020797836312xz^{31}+y^{32}-2640y^{30}z^{2}+589032y^{28}z^{4}-395654544y^{26}z^{6}-81621070740y^{24}z^{8}+25184206708848y^{22}z^{10}+8028947326999608y^{20}z^{12}+77195454038875440y^{18}z^{14}+1118897140663266966y^{16}z^{16}-80467378824098740464y^{14}z^{18}-2371007321463106703592y^{12}z^{20}-10886241598758882024624y^{10}z^{22}+72101660399250853750668y^{8}z^{24}+491746047945440400185040y^{6}z^{26}+691819795282175476492104y^{4}z^{28}+3881310382473006405861648y^{2}z^{30}+12984204345290914105535985z^{32}}{z^{4}y^{4}(y^{2}+27z^{2})^{6}(12x^{2}y^{10}+6561x^{2}y^{8}z^{2}-866052x^{2}y^{6}z^{4}-47829690x^{2}y^{4}z^{6}+3486784401x^{2}z^{10}-114xy^{10}z+48114xy^{8}z^{3}+4566456xy^{6}z^{5}-51018336xy^{4}z^{7}-2324522934xy^{2}z^{9}-6973568802xz^{11}+y^{12}-1194y^{10}z^{2}-204849y^{8}z^{4}+6219828y^{6}z^{6}+314081631y^{4}z^{8}+2324522934y^{2}z^{10}+3486784401z^{12})}$ |
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.1.13 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.1.15 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.15 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.d.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.d.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | 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.d.1.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.d.2.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.e.1.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.e.3.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.576.9-12.c.2.2 | $12$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.5-24.br.1.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.br.3.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.3.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 36.576.9-36.d.2.2 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 36.576.17-36.d.2.6 | $36$ | $3$ | $3$ | $17$ | $0$ | $1^{8}\cdot4^{2}$ |
| 36.576.17-36.h.1.2 | $36$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot4^{2}$ |
| 60.384.5-60.n.3.6 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.384.5-60.n.4.8 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.384.5-60.o.3.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.384.5-60.o.4.8 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.960.33-60.d.1.2 | $60$ | $5$ | $5$ | $33$ | $2$ | $1^{16}\cdot8^{2}$ |
| 60.1152.33-60.d.2.2 | $60$ | $6$ | $6$ | $33$ | $2$ | $1^{16}\cdot8^{2}$ |
| 60.1920.65-60.d.2.4 | $60$ | $10$ | $10$ | $65$ | $6$ | $1^{32}\cdot8^{4}$ |
| 84.384.5-84.n.2.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.n.3.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.o.1.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.o.2.7 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kj.3.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kj.4.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kq.3.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kq.4.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.n.3.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.n.4.8 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.o.3.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.o.4.8 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.n.2.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.n.3.3 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.o.1.5 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.o.2.7 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kj.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kj.4.13 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kq.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kq.4.13 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.n.3.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.n.4.8 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.o.2.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.o.4.8 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.n.2.4 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.n.3.3 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.o.1.3 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.o.2.7 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kj.2.9 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kj.3.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kq.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kq.2.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.n.3.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.n.4.8 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.o.2.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.o.4.8 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kj.2.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kj.3.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kq.2.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kq.3.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |