↑

Properties

Label 20.72.1.f.2
Level $20$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $6$

Related objects

Downloads

Learn more

Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $20$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $6$ are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $1^{6}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.72.1.9

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}1&9\\0&17\end{bmatrix}$, $\begin{bmatrix}1&18\\0&7\end{bmatrix}$, $\begin{bmatrix}1&19\\0&1\end{bmatrix}$, $\begin{bmatrix}9&16\\0&1\end{bmatrix}$, $\begin{bmatrix}11&14\\0&17\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $D_{10}.C_2^5$
Contains $-I$: yes
Quadratic refinements: 20.144.1-20.f.2.1, 20.144.1-20.f.2.2, 20.144.1-20.f.2.3, 20.144.1-20.f.2.4, 20.144.1-20.f.2.5, 20.144.1-20.f.2.6, 20.144.1-20.f.2.7, 20.144.1-20.f.2.8, 20.144.1-20.f.2.9, 20.144.1-20.f.2.10, 20.144.1-20.f.2.11, 20.144.1-20.f.2.12, 40.144.1-20.f.2.1, 40.144.1-20.f.2.2, 40.144.1-20.f.2.3, 40.144.1-20.f.2.4, 40.144.1-20.f.2.5, 40.144.1-20.f.2.6, 40.144.1-20.f.2.7, 40.144.1-20.f.2.8, 40.144.1-20.f.2.9, 40.144.1-20.f.2.10, 40.144.1-20.f.2.11, 40.144.1-20.f.2.12, 40.144.1-20.f.2.13, 40.144.1-20.f.2.14, 40.144.1-20.f.2.15, 40.144.1-20.f.2.16, 40.144.1-20.f.2.17, 40.144.1-20.f.2.18, 40.144.1-20.f.2.19, 40.144.1-20.f.2.20, 40.144.1-20.f.2.21, 40.144.1-20.f.2.22, 40.144.1-20.f.2.23, 40.144.1-20.f.2.24, 40.144.1-20.f.2.25, 40.144.1-20.f.2.26, 40.144.1-20.f.2.27, 40.144.1-20.f.2.28, 60.144.1-20.f.2.1, 60.144.1-20.f.2.2, 60.144.1-20.f.2.3, 60.144.1-20.f.2.4, 60.144.1-20.f.2.5, 60.144.1-20.f.2.6, 60.144.1-20.f.2.7, 60.144.1-20.f.2.8, 60.144.1-20.f.2.9, 60.144.1-20.f.2.10, 60.144.1-20.f.2.11, 60.144.1-20.f.2.12, 120.144.1-20.f.2.1, 120.144.1-20.f.2.2, 120.144.1-20.f.2.3, 120.144.1-20.f.2.4, 120.144.1-20.f.2.5, 120.144.1-20.f.2.6, 120.144.1-20.f.2.7, 120.144.1-20.f.2.8, 120.144.1-20.f.2.9, 120.144.1-20.f.2.10, 120.144.1-20.f.2.11, 120.144.1-20.f.2.12, 120.144.1-20.f.2.13, 120.144.1-20.f.2.14, 120.144.1-20.f.2.15, 120.144.1-20.f.2.16, 120.144.1-20.f.2.17, 120.144.1-20.f.2.18, 120.144.1-20.f.2.19, 120.144.1-20.f.2.20, 120.144.1-20.f.2.21, 120.144.1-20.f.2.22, 120.144.1-20.f.2.23, 120.144.1-20.f.2.24, 120.144.1-20.f.2.25, 120.144.1-20.f.2.26, 120.144.1-20.f.2.27, 120.144.1-20.f.2.28, 140.144.1-20.f.2.1, 140.144.1-20.f.2.2, 140.144.1-20.f.2.3, 140.144.1-20.f.2.4, 140.144.1-20.f.2.5, 140.144.1-20.f.2.6, 140.144.1-20.f.2.7, 140.144.1-20.f.2.8, 140.144.1-20.f.2.9, 140.144.1-20.f.2.10, 140.144.1-20.f.2.11, 140.144.1-20.f.2.12, 220.144.1-20.f.2.1, 220.144.1-20.f.2.2, 220.144.1-20.f.2.3, 220.144.1-20.f.2.4, 220.144.1-20.f.2.5, 220.144.1-20.f.2.6, 220.144.1-20.f.2.7, 220.144.1-20.f.2.8, 220.144.1-20.f.2.9, 220.144.1-20.f.2.10, 220.144.1-20.f.2.11, 220.144.1-20.f.2.12, 260.144.1-20.f.2.1, 260.144.1-20.f.2.2, 260.144.1-20.f.2.3, 260.144.1-20.f.2.4, 260.144.1-20.f.2.5, 260.144.1-20.f.2.6, 260.144.1-20.f.2.7, 260.144.1-20.f.2.8, 260.144.1-20.f.2.9, 260.144.1-20.f.2.10, 260.144.1-20.f.2.11, 260.144.1-20.f.2.12, 280.144.1-20.f.2.1, 280.144.1-20.f.2.2, 280.144.1-20.f.2.3, 280.144.1-20.f.2.4, 280.144.1-20.f.2.5, 280.144.1-20.f.2.6, 280.144.1-20.f.2.7, 280.144.1-20.f.2.8, 280.144.1-20.f.2.9, 280.144.1-20.f.2.10, 280.144.1-20.f.2.11, 280.144.1-20.f.2.12, 280.144.1-20.f.2.13, 280.144.1-20.f.2.14, 280.144.1-20.f.2.15, 280.144.1-20.f.2.16, 280.144.1-20.f.2.17, 280.144.1-20.f.2.18, 280.144.1-20.f.2.19, 280.144.1-20.f.2.20, 280.144.1-20.f.2.21, 280.144.1-20.f.2.22, 280.144.1-20.f.2.23, 280.144.1-20.f.2.24, 280.144.1-20.f.2.25, 280.144.1-20.f.2.26, 280.144.1-20.f.2.27, 280.144.1-20.f.2.28
Cyclic 20-isogeny field degree: $1$
Cyclic 20-torsion field degree: $4$
Full 20-torsion field degree: $640$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - x $
 Copy content Toggle raw display

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:0:1)$, $(0:1:0)$, $(-1:1:1)$, $(1:1:1)$, $(1:-1:1)$, $(-1:-1:1)$

Maps to other modular curves

$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{24x^{2}y^{22}-7096x^{2}y^{20}z^{2}+146600x^{2}y^{18}z^{4}-928872x^{2}y^{16}z^{6}+2866160x^{2}y^{14}z^{8}-5468848x^{2}y^{12}z^{10}+7343952x^{2}y^{10}z^{12}-7191120x^{2}y^{8}z^{14}+5192312x^{2}y^{6}z^{16}-2731800x^{2}y^{4}z^{18}+953736x^{2}y^{2}z^{20}-199624x^{2}z^{22}-240xy^{22}z+19488xy^{20}z^{3}-233440xy^{18}z^{5}+1106160xy^{16}z^{7}-2889120xy^{14}z^{9}+4983680xy^{12}z^{11}-6198144xy^{10}z^{13}+5692320xy^{8}z^{15}-3895280xy^{6}z^{17}+1935840xy^{4}z^{19}-644640xy^{2}z^{21}+123376xz^{23}-y^{24}+1532y^{22}z^{2}-53362y^{20}z^{4}+414860y^{18}z^{6}-1415215y^{16}z^{8}+2846072y^{14}z^{10}-3969724y^{12}z^{12}+4005240y^{10}z^{14}-2961455y^{8}z^{16}+1597580y^{6}z^{18}-568434y^{4}z^{20}+123388y^{2}z^{22}-z^{24}}{z^{6}y^{4}(y-z)^{2}(y+z)^{2}(x^{2}y^{8}-148x^{2}y^{6}z^{2}+1082x^{2}y^{4}z^{4}-2084x^{2}y^{2}z^{6}+1165x^{2}z^{8}-10xy^{8}z+270xy^{6}z^{3}-1150xy^{4}z^{5}+1610xy^{2}z^{7}-720xz^{9}+45y^{8}z^{2}-506y^{6}z^{4}+1165y^{4}z^{6}-720y^{2}z^{8})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(4)$ $4$ $12$ $12$ $0$ $0$ full Jacobian
$X_{\pm1}(5)$ $5$ $6$ $6$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\pm1}(10)$ $10$ $2$ $2$ $0$ $0$ full Jacobian
20.36.0.b.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
$X_0(20)$ $20$ $2$ $2$ $1$ $0$ 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
$X_{\pm1}(20)$ $20$ $2$ $2$ $3$ $0$ $2$
20.144.3.e.4 $20$ $2$ $2$ $3$ $0$ $2$
20.144.3.f.1 $20$ $2$ $2$ $3$ $0$ $2$
20.144.3.f.3 $20$ $2$ $2$ $3$ $0$ $2$
20.144.5.b.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.n.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.u.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.w.1 $20$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
20.360.13.i.1 $20$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
40.144.3.e.2 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.e.4 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.f.1 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.f.3 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.i.1 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.j.1 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.m.2 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.n.2 $40$ $2$ $2$ $3$ $0$ $2$
40.144.5.ba.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.dq.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.fm.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.fz.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.gm.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.gn.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.gu.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.gv.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.hc.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.hd.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.hg.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.hh.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.7.fh.2 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
40.144.7.fi.2 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
40.144.7.fl.1 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
40.144.7.fm.1 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
60.144.3.h.1 $60$ $2$ $2$ $3$ $0$ $2$
60.144.3.h.3 $60$ $2$ $2$ $3$ $0$ $2$
60.144.3.i.2 $60$ $2$ $2$ $3$ $0$ $2$
60.144.3.i.4 $60$ $2$ $2$ $3$ $0$ $2$
60.144.5.ee.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.eg.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.fs.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.fu.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.216.13.ba.2 $60$ $3$ $3$ $13$ $0$ $1^{6}\cdot2^{3}$
60.288.13.ie.2 $60$ $4$ $4$ $13$ $0$ $1^{6}\cdot2^{3}$
100.360.13.f.2 $100$ $5$ $5$ $13$ $?$ not computed
120.144.3.l.1 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.l.3 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.m.2 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.m.4 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.p.1 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.q.1 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.t.2 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.u.2 $120$ $2$ $2$ $3$ $?$ not computed
120.144.5.bdd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bdr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bpr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzs.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzt.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzw.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.bzx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.7.fpx.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.fpy.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.fqb.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.fqc.1 $120$ $2$ $2$ $7$ $?$ not computed
140.144.3.e.2 $140$ $2$ $2$ $3$ $?$ not computed
140.144.3.e.4 $140$ $2$ $2$ $3$ $?$ not computed
140.144.3.f.1 $140$ $2$ $2$ $3$ $?$ not computed
140.144.3.f.3 $140$ $2$ $2$ $3$ $?$ not computed
140.144.5.ci.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.cj.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.cq.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.cr.1 $140$ $2$ $2$ $5$ $?$ not computed
220.144.3.e.3 $220$ $2$ $2$ $3$ $?$ not computed
220.144.3.e.4 $220$ $2$ $2$ $3$ $?$ not computed
220.144.3.f.3 $220$ $2$ $2$ $3$ $?$ not computed
220.144.3.f.4 $220$ $2$ $2$ $3$ $?$ not computed
220.144.5.ci.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.cj.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.cq.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.cr.1 $220$ $2$ $2$ $5$ $?$ not computed
260.144.3.m.1 $260$ $2$ $2$ $3$ $?$ not computed
260.144.3.m.3 $260$ $2$ $2$ $3$ $?$ not computed
260.144.3.n.2 $260$ $2$ $2$ $3$ $?$ not computed
260.144.3.n.4 $260$ $2$ $2$ $3$ $?$ not computed
260.144.5.ci.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.cj.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.cq.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.cr.1 $260$ $2$ $2$ $5$ $?$ not computed
280.144.3.e.2 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.e.4 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.f.1 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.f.3 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.i.1 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.j.1 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.m.2 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.n.2 $280$ $2$ $2$ $3$ $?$ not computed
280.144.5.qe.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ql.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.si.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.sp.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tk.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tl.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.to.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tp.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.ts.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tt.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tw.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.tx.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.7.gv.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.gw.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.gz.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.ha.1 $280$ $2$ $2$ $7$ $?$ not computed