Invariants
| Level: | $120$ | $\SL_2$-level: | $15$ | Newform level: | $15$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot5^{2}\cdot15^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 15G1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}4&5\\51&53\end{bmatrix}$, $\begin{bmatrix}63&80\\92&81\end{bmatrix}$, $\begin{bmatrix}77&69\\26&115\end{bmatrix}$, $\begin{bmatrix}83&9\\47&70\end{bmatrix}$, $\begin{bmatrix}118&59\\81&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.48.1.bw.2 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $12$ |
| Cyclic 120-torsion field degree: | $384$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 15.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - 2 x z - y^{2} - z^{2} $ |
| $=$ | $17 x^{2} + 22 x z + 5 y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} + 65 x^{2} y^{2} + 410 x^{2} z^{2} + 4 y^{4} + 12 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{5}{2^3}\cdot\frac{2862251226000000xyz^{9}w-25907146704000000xyz^{7}w^{3}+50504229495360000xyz^{5}w^{5}-9666964164096000xyz^{3}w^{7}-552282791270400xyzw^{9}+637762306562500xz^{11}-10147300124750000xz^{9}w^{2}+41920976925000000xz^{7}w^{4}-38089377182800000xz^{5}w^{6}-278465368456000xz^{3}w^{8}-65822725228800xzw^{10}+715563267300000yz^{10}w-7192320912900000yz^{8}w^{3}+17671391840232000yz^{6}w^{5}-7842991057545600yz^{4}w^{7}-296306055475200yz^{2}w^{9}-9471147969024yw^{11}+159440544640625z^{12}-2696264955028125z^{10}w^{2}+12698300313956250z^{8}w^{4}-16201138796356000z^{6}w^{6}+2408948618538600z^{4}w^{8}+191799673223760z^{2}w^{10}+1138274232736w^{12}}{210600000xyz^{9}w-10663520000xyz^{7}w^{3}+93406316800xyz^{5}w^{5}-89236561920xyz^{3}w^{7}-3643952128xyzw^{9}-9112500xz^{11}+1599750000xz^{9}w^{2}-33438000000xz^{7}w^{4}+123476208000xz^{5}w^{6}-26214590400xz^{3}w^{8}-471614208xzw^{10}+52650000yz^{10}w-2692930000yz^{8}w^{3}+25630559200yz^{6}w^{5}-39316559680yz^{4}w^{7}+1018498048yz^{2}w^{9}-8279808yw^{11}-2278125z^{12}+402215625z^{10}w^{2}-8658881250z^{8}w^{4}+38536502000z^{6}w^{6}-22367466600z^{4}w^{8}-1260580752z^{2}w^{10}-12713888w^{12}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.bw.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 225X^{4}+65X^{2}Y^{2}+4Y^{4}+410X^{2}Z^{2}+12Y^{2}Z^{2}+9Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 120.48.1-15.a.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1-15.a.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.5-60.a.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.c.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.s.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.u.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bo.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bq.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bu.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bw.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.ke.2.7 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.288.5-60.oc.2.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.384.13-60.bh.2.5 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 120.480.9-60.a.1.9 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.192.5-120.em.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.es.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ia.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ig.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ng.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.nm.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.oe.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ok.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |