Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot10^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.36.1.10 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}9&16\\14&17\end{bmatrix}$, $\begin{bmatrix}13&2\\11&17\end{bmatrix}$, $\begin{bmatrix}17&14\\17&15\end{bmatrix}$, $\begin{bmatrix}19&8\\19&15\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $(C_2\times D_{20}):C_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $16$ |
| Full 20-torsion field degree: | $1280$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} + 92x - 312 $ |
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.
| Weierstrass model |
|---|
| $(3:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{5}\cdot\frac{3510x^{2}y^{10}-108000918750x^{2}y^{8}z^{2}-569788753125000x^{2}y^{6}z^{4}-574936139003906250x^{2}y^{4}z^{6}-171921563820800781250x^{2}y^{2}z^{8}-9638282574615478515625x^{2}z^{10}-4182435xy^{10}z+1665371737500xy^{8}z^{3}+1299957188671875xy^{6}z^{5}-2401000215000000000xy^{4}z^{7}-2145414835880126953125xy^{2}z^{9}-407810530298461914062500xz^{11}-y^{12}+1782688215y^{10}z^{2}+12156960150000y^{8}z^{4}+36449258383984375y^{6}z^{6}+31913031285927734375y^{4}z^{8}+10591073298458251953125y^{2}z^{10}+1193760812239990234375000z^{12}}{x^{2}y^{10}+860000x^{2}y^{8}z^{2}-432000000x^{2}y^{6}z^{4}+7731000000000x^{2}y^{4}z^{6}-12955000000000000x^{2}y^{2}z^{8}-1007750000000000000x^{2}z^{10}+204xy^{10}z+23890000xy^{8}z^{3}-85428000000xy^{6}z^{5}+146796500000000xy^{4}z^{7}+36752500000000000xy^{2}z^{9}-46769125000000000000xz^{11}+17904y^{10}z^{2}+362990000y^{8}z^{4}-618728000000y^{6}z^{6}-388068500000000y^{4}z^{8}+428862500000000000y^{2}z^{10}+149377125000000000000z^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.6.0.a.1 | $20$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 20.12.1.b.1 | $20$ | $3$ | $3$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 20.72.1.i.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.72.1.i.2 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.72.1.j.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.72.1.j.2 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.72.3.n.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.72.3.p.1 | $20$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 20.72.3.w.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.72.3.w.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.72.3.x.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.72.3.x.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.72.3.bd.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.72.3.bf.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.180.7.p.1 | $20$ | $5$ | $5$ | $7$ | $1$ | $1^{6}$ |
| 40.72.1.be.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.be.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bh.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.1.bh.2 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.72.3.bo.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 40.72.3.bu.1 | $40$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 40.72.3.da.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.da.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dd.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dd.2 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.72.3.dv.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.72.3.eb.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.72.1.by.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.by.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.bz.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.bz.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.3.kr.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.kt.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.qp.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.72.3.qp.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.72.3.qq.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.72.3.qq.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.72.3.rn.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.72.3.rp.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.108.7.g.1 | $60$ | $3$ | $3$ | $7$ | $0$ | $1^{6}$ |
| 60.144.7.lv.1 | $60$ | $4$ | $4$ | $7$ | $1$ | $1^{6}$ |
| 100.180.7.d.1 | $100$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 120.72.1.gs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gs.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.gv.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.cyy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cze.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ecy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ecy.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.edb.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.efx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.1.k.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.72.1.k.2 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.72.1.l.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.72.1.l.2 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.72.3.ba.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bb.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bi.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bi.2 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bj.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bj.2 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.bq.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.72.3.br.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.288.19.h.1 | $140$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 220.72.1.i.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.72.1.i.2 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.72.1.j.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.72.1.j.2 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.72.3.ba.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bb.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bi.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bi.2 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bj.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bj.2 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.bq.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.72.3.br.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.1.k.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.72.1.k.2 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.72.1.l.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.72.1.l.2 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.72.3.ba.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bb.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bi.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bi.2 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bj.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bj.2 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.bq.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.72.3.br.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.1.bi.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bi.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.1.bl.2 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.72.3.dm.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.dp.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ek.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.ek.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.en.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.en.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fi.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.72.3.fl.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |