Invariants
| Level: | $15$ | $\SL_2$-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: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 15G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 15.96.1.5 |
Level structure
| $\GL_2(\Z/15\Z)$-generators: | $\begin{bmatrix}1&10\\0&2\end{bmatrix}$, $\begin{bmatrix}4&11\\0&13\end{bmatrix}$, $\begin{bmatrix}14&8\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/15\Z)$-subgroup: | $D_6\times F_5$ |
| Contains $-I$: | no $\quad$ (see 15.48.1.a.1 for the level structure with $-I$) |
| Cyclic 15-isogeny field degree: | $1$ |
| Cyclic 15-torsion field degree: | $4$ |
| Full 15-torsion field degree: | $240$ |
Jacobian
| Conductor: | $3\cdot5$ |
| 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 has rational points, including 4 rational_cusps 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 | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(1:-1:1)$, $(0:1:0)$, $(0:-2:1)$, $(0:1:1)$ | ||
| 50.a2 | no | $\tfrac{-121945}{32}$ | $= -1 \cdot 2^{-5} \cdot 5 \cdot 29^{3}$ | $11.711$ | $(2:-4:1)$, $(2:1:1)$ |
| 50.a4 | no | $\tfrac{46969655}{32768}$ | $= 2^{-15} \cdot 5 \cdot 211^{3}$ | $17.665$ | $(-3:1:1)$, $(3/4:-7/8:1)$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 15.24.0-5.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 |
|---|---|---|---|---|---|---|
| 15.192.1-15.a.3.4 | $15$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 15.192.1-15.a.4.2 | $15$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 15.192.1-15.b.1.2 | $15$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 15.192.1-15.b.2.1 | $15$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 15.288.5-15.a.1.6 | $15$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 15.480.9-15.a.1.4 | $15$ | $5$ | $5$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 30.192.5-30.a.2.5 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.c.1.7 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.e.2.6 | $30$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.g.1.5 | $30$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 30.192.5-30.i.1.7 | $30$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 30.192.5-30.i.3.6 | $30$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 30.192.5-30.j.2.6 | $30$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 30.192.5-30.j.4.4 | $30$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 30.288.5-30.a.1.14 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 45.288.5-45.a.1.4 | $45$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 45.288.9-45.a.1.3 | $45$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot4$ |
| 45.288.9-45.c.2.6 | $45$ | $3$ | $3$ | $9$ | $0$ | $2^{2}\cdot4$ |
| 60.192.1-60.q.3.15 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.q.4.15 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.r.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.1-60.r.2.7 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.192.5-60.b.2.19 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-60.t.2.7 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-60.bp.2.11 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.192.5-60.bv.2.7 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.192.5-60.ca.1.13 | $60$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 60.192.5-60.ca.3.10 | $60$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 60.192.5-60.cb.2.9 | $60$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 60.192.5-60.cb.4.3 | $60$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 60.384.13-60.bg.2.3 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 75.480.9-75.a.1.3 | $75$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 105.192.1-105.a.1.3 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.192.1-105.a.4.6 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.192.1-105.b.2.2 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.192.1-105.b.3.7 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.to.3.27 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.to.4.23 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tp.3.27 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tp.4.23 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tq.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tq.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tr.1.13 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.tr.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.5-120.ej.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ep.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.hx.2.18 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.id.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.nd.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.nj.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ob.2.19 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.oh.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wm.1.26 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wm.2.20 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wn.1.22 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wn.2.20 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wo.2.19 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wo.4.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wp.2.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.wp.4.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 165.192.1-165.a.1.8 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.192.1-165.a.2.8 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.192.1-165.b.1.4 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.192.1-165.b.2.4 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 195.192.1-195.a.2.2 | $195$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 195.192.1-195.a.3.6 | $195$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 195.192.1-195.b.1.3 | $195$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 195.192.1-195.b.4.4 | $195$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.192.5-210.i.2.10 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.j.2.8 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.m.2.10 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.n.2.7 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.q.2.16 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.q.4.16 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.r.2.16 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.192.5-210.r.4.16 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 255.192.1-255.a.1.6 | $255$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 255.192.1-255.a.3.4 | $255$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 255.192.1-255.b.1.3 | $255$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 255.192.1-255.b.2.2 | $255$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 285.192.1-285.a.1.8 | $285$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 285.192.1-285.a.2.8 | $285$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 285.192.1-285.b.1.8 | $285$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 285.192.1-285.b.2.8 | $285$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.192.5-330.i.2.14 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.j.2.10 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.m.2.14 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.n.2.14 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.q.1.16 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.q.2.16 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.r.3.14 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.192.5-330.r.4.8 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |