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: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot5^{2}\cdot15^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 15G1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}1&35\\12&89\end{bmatrix}$, $\begin{bmatrix}8&1\\95&99\end{bmatrix}$, $\begin{bmatrix}41&43\\38&51\end{bmatrix}$, $\begin{bmatrix}41&55\\42&109\end{bmatrix}$, $\begin{bmatrix}67&62\\48&41\end{bmatrix}$, $\begin{bmatrix}95&6\\44&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 15.48.1.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $12$ |
| Cyclic 120-torsion field degree: | $192$ |
| Full 120-torsion field degree: | $368640$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 15.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + \left(x + 1\right) y $ | $=$ | $ x^{3} + x^{2} - 5x + 2 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{105x^{2}y^{17}-2909x^{2}y^{16}z+10468x^{2}y^{15}z^{2}+351900x^{2}y^{14}z^{3}-3640647x^{2}y^{13}z^{4}-22636290x^{2}y^{12}z^{5}+177699243x^{2}y^{11}z^{6}+737564202x^{2}y^{10}z^{7}-4070427135x^{2}y^{9}z^{8}-15564596259x^{2}y^{8}z^{9}+44025093400x^{2}y^{7}z^{10}+195751036150x^{2}y^{6}z^{11}-162898127679x^{2}y^{5}z^{12}-1294888654280x^{2}y^{4}z^{13}-662054425877x^{2}y^{3}z^{14}+3187144498470x^{2}y^{2}z^{15}+5047660836832x^{2}yz^{16}+2224217422129x^{2}z^{17}+17xy^{18}-216xy^{17}z+10515xy^{16}z^{2}-179081xy^{15}z^{3}-1481250xy^{14}z^{4}+14656177xy^{13}z^{5}+57080906xy^{12}z^{6}-606125820xy^{11}z^{7}-1771146819xy^{10}z^{8}+11865210460xy^{9}z^{9}+35712130935xy^{8}z^{10}-117648244571xy^{7}z^{11}-435153006116xy^{6}z^{12}+435705757968xy^{5}z^{13}+2787877544530xy^{4}z^{14}+1197136969230xy^{3}z^{15}-6672040631433xy^{2}z^{16}-10056788345855xyz^{17}-4304822312129xz^{18}+y^{19}+216y^{18}z-1476y^{17}z^{2}-92627y^{16}z^{3}+439393y^{15}z^{4}+6284608y^{14}z^{5}-34566316y^{13}z^{6}-261470739y^{12}z^{7}+875261295y^{11}z^{8}+6079310900y^{10}z^{9}-9117786976y^{9}z^{10}-81345748049y^{8}z^{11}+6242429357y^{7}z^{12}+572964102846y^{6}z^{13}+556805814469y^{5}z^{14}-1613864040618y^{4}z^{15}-3182122533292y^{3}z^{16}-386364060818y^{2}z^{17}+2614709172272yz^{18}+1515427487741z^{19}}{z^{5}(579x^{2}y^{11}z-5788x^{2}y^{10}z^{2}-74845x^{2}y^{9}z^{3}+447402x^{2}y^{8}z^{4}+3501550x^{2}y^{7}z^{5}-10782902x^{2}y^{6}z^{6}-78028524x^{2}y^{5}z^{7}+79870890x^{2}y^{4}z^{8}+820546338x^{2}y^{3}z^{9}+253690020x^{2}y^{2}z^{10}-3259172032x^{2}yz^{11}-3781962688x^{2}z^{12}+7xy^{13}-131xy^{12}z-3018xy^{11}z^{2}+31287xy^{10}z^{3}+243575xy^{9}z^{4}-1690959xy^{8}z^{5}-9460100xy^{7}z^{6}+33800918xy^{6}z^{7}+187873494xy^{5}z^{8}-238414425xy^{4}z^{9}-1822089348xy^{3}z^{10}-322653564xy^{2}z^{11}+6796559400xyz^{12}+7319732864xz^{13}+y^{14}-84y^{13}z+583y^{12}z^{2}+16929y^{11}z^{3}-72136y^{10}z^{4}-1033119y^{9}z^{5}+1750960y^{8}z^{6}+26845156y^{7}z^{7}-1411944y^{6}z^{8}-316925922y^{5}z^{9}-318472932y^{4}z^{10}+1455951164y^{3}z^{11}+2548889616y^{2}z^{12}-818771200yz^{13}-2576767072z^{14})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\pm1}(5)$ | $5$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.8.0-3.a.1.2 | $24$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
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.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.192.1-15.a.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-15.a.4.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-15.b.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-15.b.2.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.q.3.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.q.4.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.r.1.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.r.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.to.3.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.to.4.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tp.3.19 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tp.4.23 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tq.1.17 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tq.2.19 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tr.1.25 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tr.2.27 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.5-30.a.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.b.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.c.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.e.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.g.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.i.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.i.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.j.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-30.j.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.t.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bp.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.bv.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.ca.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.ca.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.cb.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-60.cb.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ej.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ep.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.hx.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.id.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.nd.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.nj.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ob.2.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.oh.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wm.1.23 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wm.2.31 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wn.1.19 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wn.2.23 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wo.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wo.4.30 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wp.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wp.4.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-15.a.1.1 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.288.5-30.a.1.19 | $120$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.384.13-60.bg.2.10 | $120$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 120.480.9-15.a.1.9 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |