Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $1 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot12$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $1$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 12A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.32.1.2 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}5&54\\18&7\end{bmatrix}$, $\begin{bmatrix}10&29\\51&49\end{bmatrix}$, $\begin{bmatrix}32&13\\45&4\end{bmatrix}$, $\begin{bmatrix}59&5\\3&38\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.16.1.a.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $36$ |
| Cyclic 60-torsion field degree: | $576$ |
| Full 60-torsion field degree: | $69120$ |
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} + 16x + 180 $ |
Rational points
This modular curve has rational points, including 2 rational cusps, 2 rational CM points and 2 known non-cuspidal non-CM 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 | |
|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(-2:-12:1)$, $(-2:12:1)$, $(22:108:1)$ | |
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(4:18:1)$ | ||
| 162.a1 | no | $\tfrac{-35937}{4}$ | $= -1 \cdot 2^{-2} \cdot 3^{3} \cdot 11^{3}$ | $10.490$ | $(4:-18:1)$ |
| 162.a2 | no | $\tfrac{109503}{64}$ | $= 2^{-6} \cdot 3^{2} \cdot 23^{3}$ | $11.604$ | $(-5:0:1)$ |
| 27.a1 | $-27$ | $-12288000$ | $= -1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ | $16.324$ | $(22:-108:1)$ |
Maps to other modular curves
$j$-invariant map of degree 16 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{12x^{2}y^{5}+738x^{2}y^{4}z-272x^{2}y^{3}z^{2}-138570x^{2}y^{2}z^{3}-183732x^{2}yz^{4}+9525103x^{2}z^{5}-xy^{6}-152xy^{5}z-3712xy^{4}z^{2}+39628xy^{3}z^{3}+876775xy^{2}z^{4}-2410800xyz^{5}-39542204xz^{6}-29y^{6}z+292y^{5}z^{2}+20367y^{4}z^{3}-106796y^{3}z^{4}-4335647y^{2}z^{5}+9507504yz^{6}+282610588z^{7}}{z^{4}(97x^{2}z+xy^{2}-164xz^{2}-17y^{2}z+3316z^{3})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(4)$ | $4$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
| 15.8.0-3.a.1.2 | $15$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.8.0-3.a.1.2 | $15$ | $4$ | $4$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.64.1-12.a.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-60.a.1.5 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-12.b.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-60.b.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-12.c.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-60.c.1.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-12.d.1.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.64.1-60.d.1.3 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.96.2-12.c.1.5 | $60$ | $3$ | $3$ | $2$ | $0$ | $1$ |
| 60.96.3-12.o.1.1 | $60$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 60.160.6-60.a.1.7 | $60$ | $5$ | $5$ | $6$ | $2$ | $1^{5}$ |
| 60.192.7-60.g.1.27 | $60$ | $6$ | $6$ | $7$ | $1$ | $1^{6}$ |
| 60.320.12-60.a.1.28 | $60$ | $10$ | $10$ | $12$ | $5$ | $1^{11}$ |
| 120.64.1-24.a.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-120.a.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-24.b.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-120.b.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-24.c.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-120.c.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-24.d.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.64.1-120.d.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.128.4-24.a.1.16 | $120$ | $4$ | $4$ | $4$ | $?$ | not computed |
| 180.96.3-36.c.1.7 | $180$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 180.96.3-36.d.1.4 | $180$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 180.96.4-36.c.1.8 | $180$ | $3$ | $3$ | $4$ | $?$ | not computed |