Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $80$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.96.3.25 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}1&4\\9&7\end{bmatrix}$, $\begin{bmatrix}2&1\\7&18\end{bmatrix}$, $\begin{bmatrix}5&14\\18&19\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_3:D_4\times F_5$ |
| Contains $-I$: | no $\quad$ (see 20.48.3.i.2 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $6$ |
| Cyclic 20-torsion field degree: | $48$ |
| Full 20-torsion field degree: | $480$ |
Jacobian
| Conductor: | $2^{12}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 80.2.a.a, 80.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 6 x^{3} y - 22 x^{2} y^{2} + 12 x^{2} y z - 3 x^{2} z^{2} + 6 x y^{3} + 14 x y^{2} z - 2 x y z^{2} + \cdots + 2 z^{4} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:1:0)$, $(1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{944784x^{12}-5668704x^{10}z^{2}+11337408x^{9}z^{3}-185650056x^{8}z^{4}+717091056x^{7}z^{5}-1844218368x^{6}z^{6}+5339919168x^{5}z^{7}-8537245371x^{4}z^{8}-10075176576x^{3}z^{9}-4900920579440x^{2}y^{10}+46367798662032x^{2}y^{9}z-177273459133152x^{2}y^{8}z^{2}+346505346535104x^{2}y^{7}z^{3}-354754307358468x^{2}y^{6}z^{4}+158497749221184x^{2}y^{5}z^{5}+11336353272864x^{2}y^{4}z^{6}-35170697402856x^{2}y^{3}z^{7}+10719376046766x^{2}y^{2}z^{8}-1402682646024x^{2}yz^{9}+145875948678x^{2}z^{10}+1454232130032xy^{11}-9564722394272xy^{10}z+16047565483280xy^{9}z^{2}+21615671944440xy^{8}z^{3}-97582592311704xy^{7}z^{4}+99193866658992xy^{6}z^{5}-13345762687488xy^{5}z^{6}-27090678079800xy^{4}z^{7}+7487991343008xy^{3}z^{8}+1951242893964xy^{2}z^{9}-36952789356xyz^{10}-157004625024xz^{11}+944784y^{12}-1454243467440y^{11}z+13738583595736y^{10}z^{2}-54913593066632y^{9}z^{3}+123906691539976y^{8}z^{4}-178736568208200y^{7}z^{5}+171289433031072y^{6}z^{6}-96251223849984y^{5}z^{7}+11724073442289y^{4}z^{8}+18597622482252y^{3}z^{9}-8910588624678y^{2}z^{10}+969344773020yz^{11}+38682376200z^{12}}{944784x^{8}z^{4}-3779136x^{7}z^{5}+9447840x^{6}z^{6}-26453952x^{5}z^{7}+99202320x^{4}z^{8}-409091472x^{3}z^{9}+18767872000x^{2}y^{10}-122485800960x^{2}y^{9}z+353151713280x^{2}y^{8}z^{2}-608551206912x^{2}y^{7}z^{3}+714937059072x^{2}y^{6}z^{4}-612965528784x^{2}y^{5}z^{5}+394783631136x^{2}y^{4}z^{6}-190953334656x^{2}y^{3}z^{7}+67000659516x^{2}y^{2}z^{8}-14602239360x^{2}yz^{9}+1691163360x^{2}z^{10}-5568921600xy^{11}+20280770560xy^{10}z-11902320640xy^{9}z^{2}-52414906368xy^{8}z^{3}+131964093936xy^{7}z^{4}-159762405600xy^{6}z^{5}+124706505744xy^{5}z^{6}-65888852760xy^{4}z^{7}+22096315944xy^{3}z^{8}-3202054416xy^{2}z^{9}-1524580056xyz^{10}+285626088xz^{11}+5568921600y^{11}z-36264181760y^{10}z^{2}+113833246720y^{9}z^{3}-234478802288y^{8}z^{4}+350830251984y^{7}z^{5}-394749478776y^{6}z^{6}+340326597384y^{5}z^{7}-226825715340y^{4}z^{8}+115920673992y^{3}z^{9}-43726804560y^{2}z^{10}+11159130240yz^{11}-1174500567z^{12}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.24.0-5.a.2.1 | $20$ | $4$ | $4$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 20.192.5-20.a.2.4 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.192.5-20.c.2.5 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.192.5-20.e.1.2 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.192.5-20.g.1.4 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.288.7-20.bd.1.1 | $20$ | $3$ | $3$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 20.480.15-20.bj.1.2 | $20$ | $5$ | $5$ | $15$ | $2$ | $1^{6}\cdot2^{3}$ |
| 40.192.5-40.b.2.3 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.192.5-40.h.2.6 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.192.5-40.n.2.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.192.5-40.t.1.8 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 40.384.13-40.q.2.12 | $40$ | $4$ | $4$ | $13$ | $1$ | $1^{4}\cdot2\cdot4$ |
| 60.192.5-60.v.1.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-60.x.1.5 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.192.5-60.cg.1.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.192.5-60.ci.1.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.288.11-60.ba.1.2 | $60$ | $3$ | $3$ | $11$ | $3$ | $1^{2}\cdot2^{3}$ |
| 60.384.13-60.bg.1.2 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{2}$ |
| 100.480.15-100.i.2.2 | $100$ | $5$ | $5$ | $15$ | $?$ | not computed |
| 120.192.5-120.ij.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ip.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.ber.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.bex.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.192.5-140.bs.2.6 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.192.5-140.bt.1.8 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.192.5-140.bw.1.6 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.192.5-140.bx.1.4 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.192.5-220.i.2.2 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.192.5-220.j.1.5 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.192.5-220.m.1.3 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.192.5-220.n.1.4 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.192.5-260.i.1.6 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.192.5-260.j.1.4 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.192.5-260.m.2.6 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.192.5-260.n.1.8 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.5-280.fk.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.5-280.fn.1.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.5-280.fw.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.192.5-280.fz.1.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |