Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| 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 | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.234 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&40\\13&57\end{bmatrix}$, $\begin{bmatrix}29&56\\16&17\end{bmatrix}$, $\begin{bmatrix}37&4\\55&23\end{bmatrix}$, $\begin{bmatrix}45&28\\28&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.j.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 15x + 22 $ |
Rational points
This modular curve has 2 rational CM points but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(2:0:1)$, $(0:1:0)$ |
| 32.a1 | $-16$ | $287496$ | $= 2^{3} \cdot 3^{3} \cdot 11^{3}$ | $12.569$ | $(-1:-6:1)$, $(-1:6:1)$, $(3:-2:1)$, $(3:2:1)$ |
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 |
|---|---|---|---|---|---|---|
| 60.36.1-12.c.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1-12.c.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.144.3-12.d.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bj.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bp.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.br.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fl.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fn.1.2 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.ft.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.fv.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.360.13-60.q.1.1 | $60$ | $5$ | $5$ | $13$ | $9$ | $1^{12}$ |
| 60.432.13-60.bj.1.15 | $60$ | $6$ | $6$ | $13$ | $3$ | $1^{12}$ |
| 60.720.25-60.cu.1.13 | $60$ | $10$ | $10$ | $25$ | $17$ | $1^{24}$ |
| 120.144.3-24.cb.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.iy.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.km.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.la.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.no.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.no.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.np.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.np.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nw.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nw.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nx.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nx.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oi.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oi.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oj.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oj.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.om.1.8 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.om.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.on.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.on.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bii.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.biw.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bkm.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bla.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byy.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byy.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byz.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byz.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzc.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzc.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzd.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzd.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzo.1.5 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzo.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzp.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzp.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzs.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzs.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzt.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzt.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.216.7-36.j.1.3 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |