Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
| Index: | $20$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $1 = 1 + \frac{ 20 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $10^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.20.1.2 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}9&1\\6&7\end{bmatrix}$, $\begin{bmatrix}9&6\\1&1\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_{12}.D_6$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $18$ |
| Cyclic 10-torsion field degree: | $72$ |
| Full 10-torsion field degree: | $144$ |
Jacobian
| Conductor: | $2^{2}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x z + 2 x w + y z - y w $ |
| $=$ | $8 x^{2} + 12 x y + 7 y^{2} + z^{2} + 3 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} + 78 x^{3} z + 35 x^{2} y^{2} + 49 x^{2} z^{2} + 40 x y^{2} z + 12 x z^{3} + 15 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Maps to other modular curves
$j$-invariant map of degree 20 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot7\,\frac{472500xy^{5}+382000xy^{3}w^{2}+423500xyw^{4}-283500y^{6}-546875y^{4}w^{2}-263750y^{2}w^{4}-121784z^{6}-317516z^{5}w-25145z^{4}w^{2}+66970z^{3}w^{3}-65670z^{2}w^{4}-34956zw^{5}-1899w^{6}}{122500xy^{5}-63000xy^{3}w^{2}-36500xyw^{4}-73500y^{6}+153125y^{4}w^{2}-70000y^{2}w^{4}-1624z^{6}-6972z^{5}w-7329z^{4}w^{2}-9014z^{3}w^{3}-31719z^{2}w^{4}-27350zw^{5}-6217w^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.2.0.a.1 | $10$ | $10$ | $10$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 10.40.1.c.1 | $10$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 10.40.1.d.1 | $10$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 10.60.3.e.1 | $10$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 10.60.3.f.1 | $10$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 20.40.1.g.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.40.1.j.1 | $20$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 20.80.5.c.1 | $20$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
| 30.40.1.g.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.40.1.j.1 | $30$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 30.60.5.s.1 | $30$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 30.80.5.c.1 | $30$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.40.1.y.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.bb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.bk.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.40.1.bn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 50.100.5.b.1 | $50$ | $5$ | $5$ | $5$ | $4$ | $2^{2}$ |
| 50.500.37.b.1 | $50$ | $25$ | $25$ | $37$ | $24$ | $2^{4}\cdot6^{2}\cdot8^{2}$ |
| 60.40.1.s.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.40.1.bb.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 70.40.1.c.1 | $70$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 70.40.1.d.1 | $70$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 70.160.11.b.1 | $70$ | $8$ | $8$ | $11$ | $2$ | $1^{6}\cdot2^{2}$ |
| 70.420.33.h.1 | $70$ | $21$ | $21$ | $33$ | $22$ | $1^{4}\cdot2^{7}\cdot3^{2}\cdot4^{2}$ |
| 70.560.43.b.1 | $70$ | $28$ | $28$ | $43$ | $24$ | $1^{10}\cdot2^{9}\cdot3^{2}\cdot4^{2}$ |
| 110.40.1.c.1 | $110$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 110.40.1.d.1 | $110$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 110.240.19.b.1 | $110$ | $12$ | $12$ | $19$ | $?$ | not computed |
| 120.40.1.cu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.cx.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.ee.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.40.1.eh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 130.40.1.c.1 | $130$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 130.40.1.d.1 | $130$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 130.280.21.b.1 | $130$ | $14$ | $14$ | $21$ | $?$ | not computed |
| 140.40.1.g.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 140.40.1.j.1 | $140$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 170.40.1.c.1 | $170$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 170.40.1.d.1 | $170$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 190.40.1.c.1 | $190$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 190.40.1.d.1 | $190$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.40.1.g.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 210.40.1.j.1 | $210$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.40.1.g.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 220.40.1.j.1 | $220$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 230.40.1.c.1 | $230$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 230.40.1.d.1 | $230$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.40.1.g.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 260.40.1.j.1 | $260$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.y.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.bb.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.bk.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.40.1.bn.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 290.40.1.c.1 | $290$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 290.40.1.d.1 | $290$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 310.40.1.c.1 | $310$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 310.40.1.d.1 | $310$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.40.1.g.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 330.40.1.j.1 | $330$ | $2$ | $2$ | $1$ | $?$ | dimension zero |