Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $80$ | ||
| Index: | $30$ | $\PSL_2$-index: | $30$ | ||||
| Genus: | $1 = 1 + \frac{ 30 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $10^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 10E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.30.1.1 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}7&13\\6&13\end{bmatrix}$, $\begin{bmatrix}9&19\\7&16\end{bmatrix}$, $\begin{bmatrix}11&7\\13&14\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $12$ |
| Cyclic 20-torsion field degree: | $96$ |
| Full 20-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - x $ |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational 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 | $\infty$ | $0.000$ | $(0:1:0)$ | ||
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 30 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{5x^{2}y^{8}+60x^{2}y^{6}z^{2}+1878x^{2}y^{4}z^{4}-89716x^{2}y^{2}z^{6}+76397x^{2}z^{8}-15xy^{8}z-788xy^{6}z^{3}-10434xy^{4}z^{5}+216444xy^{2}z^{7}-232647xz^{9}+y^{10}-20y^{8}z^{2}+1246y^{6}z^{4}+48700y^{4}z^{6}-208887y^{2}z^{8}+1728z^{10}}{z^{3}(x^{2}y^{4}z+3x^{2}y^{2}z^{3}-x^{2}z^{5}-xy^{6}-3xy^{4}z^{2}-2xy^{2}z^{4}+xz^{6}+2y^{6}z-4y^{2}z^{5}+z^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 20.60.3.d.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.60.3.e.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.60.3.p.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.60.3.q.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 20.90.3.f.1 | $20$ | $3$ | $3$ | $3$ | $1$ | $1^{2}$ |
| 20.120.6.e.1 | $20$ | $4$ | $4$ | $6$ | $2$ | $1^{5}$ |
| 40.60.3.j.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.m.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.bu.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.60.3.ca.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.s.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.t.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.bn.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.60.3.bo.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.90.4.z.1 | $60$ | $3$ | $3$ | $4$ | $2$ | $1^{3}$ |
| 60.120.8.bt.1 | $60$ | $4$ | $4$ | $8$ | $2$ | $1^{7}$ |
| 100.150.9.b.1 | $100$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.60.3.co.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.cu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.fm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.60.3.fs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.60.3.bk.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.60.3.bl.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.60.3.bn.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.60.3.bo.1 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.240.18.bb.1 | $140$ | $8$ | $8$ | $18$ | $?$ | not computed |
| 220.60.3.bk.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.60.3.bl.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.60.3.bn.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.60.3.bo.1 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.60.3.bk.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.60.3.bl.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.60.3.bn.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.60.3.bo.1 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.fa.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.fg.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.fm.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.60.3.fs.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |