Invariants
| Level: | $120$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 36$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}3&86\\118&15\end{bmatrix}$, $\begin{bmatrix}50&37\\23&116\end{bmatrix}$, $\begin{bmatrix}62&107\\61&68\end{bmatrix}$, $\begin{bmatrix}65&106\\86&89\end{bmatrix}$, $\begin{bmatrix}79&52\\88&53\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.h.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $491520$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 15x - 22 $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{126x^{2}y^{10}+4183542x^{2}y^{8}z^{2}+6594451704x^{2}y^{6}z^{4}+2036393704458x^{2}y^{4}z^{6}+192026385654390x^{2}y^{2}z^{8}+5381361959442849x^{2}z^{10}+6759xy^{10}z+62716572xy^{8}z^{3}+55655794413xy^{6}z^{5}+12576103833960xy^{4}z^{7}+977386233246345xy^{2}z^{9}+24022946641071294xz^{11}+y^{12}+206334y^{10}z^{2}+688918554y^{8}z^{4}+326484651510y^{6}z^{6}+43837489662075y^{4}z^{8}+2019166524778266y^{2}z^{10}+26520445444351509z^{12}}{18x^{2}y^{10}-1242x^{2}y^{8}z^{2}-4536x^{2}y^{6}z^{4}-10206x^{2}y^{4}z^{6}-4374x^{2}y^{2}z^{8}+6561x^{2}z^{10}+63xy^{10}z+4104xy^{8}z^{3}+11421xy^{6}z^{5}+20412xy^{4}z^{7}+6561xy^{2}z^{9}-13122xz^{11}+y^{12}-486y^{10}z^{2}+7074y^{8}z^{4}+40986y^{6}z^{6}+112995y^{4}z^{8}+56862y^{2}z^{10}-72171z^{12}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.36.1-12.b.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.36.1-12.b.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.144.3-12.c.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-12.v.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-12.bl.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-12.bn.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.bz.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fh.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fj.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.fm.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fp.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-60.fr.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jk.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.jy.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bhg.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bhu.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjk.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bjy.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.360.13-60.o.1.3 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.432.13-60.bh.1.26 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |