Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot10^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 10D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.48.1.1 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}4&1\\1&3\end{bmatrix}$, $\begin{bmatrix}8&9\\7&5\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_3\times F_5$ |
| Contains $-I$: | no $\quad$ (see 10.24.1.a.1 for the level structure with $-I$) |
| Cyclic 10-isogeny field degree: | $3$ |
| Cyclic 10-torsion field degree: | $3$ |
| Full 10-torsion field degree: | $60$ |
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} - 41x - 116 $ |
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 |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{8x^{2}y^{6}+1560x^{2}y^{4}z^{2}+94264x^{2}y^{2}z^{4}+1875624x^{2}z^{6}+80xy^{6}z+13392xy^{4}z^{3}+777584xy^{2}z^{5}+15174128xz^{7}+y^{8}+332y^{6}z^{2}+37030y^{4}z^{4}+1771212y^{2}z^{6}+30686529z^{8}}{z^{3}y^{2}(122x^{2}z+xy^{2}+987xz^{2}+15y^{2}z+1996z^{3})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| $X_1(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.24.0-5.a.1.1 | $10$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| $X_1(2,10)$ | $10$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 10.240.5-10.c.1.2 | $10$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.96.3-20.a.2.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.96.3-20.c.1.3 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.192.5-20.a.1.3 | $20$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.144.5-30.a.2.1 | $30$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 30.192.5-30.a.2.1 | $30$ | $4$ | $4$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.96.3-40.a.2.1 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.96.3-40.c.1.5 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 50.240.5-50.a.1.2 | $50$ | $5$ | $5$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.96.3-60.d.1.7 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.96.3-60.f.2.5 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 70.144.1-70.b.2.4 | $70$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 70.384.13-70.a.2.2 | $70$ | $8$ | $8$ | $13$ | $0$ | $1^{4}\cdot2^{4}$ |
| 70.1008.37-70.a.1.2 | $70$ | $21$ | $21$ | $37$ | $4$ | $1^{4}\cdot2^{8}\cdot4^{4}$ |
| 70.1344.49-70.a.1.2 | $70$ | $28$ | $28$ | $49$ | $4$ | $1^{8}\cdot2^{12}\cdot4^{4}$ |
| 90.144.1-90.a.1.4 | $90$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 120.96.3-120.e.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3-120.g.2.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 130.144.1-130.b.1.4 | $130$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 140.96.3-140.a.2.5 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.96.3-140.c.1.7 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 190.144.1-190.b.1.4 | $190$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 220.96.3-220.a.1.7 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.96.3-220.c.1.7 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.96.3-260.a.1.7 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.96.3-260.c.2.5 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3-280.a.2.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3-280.c.1.9 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 310.144.1-310.b.1.4 | $310$ | $3$ | $3$ | $1$ | $?$ | dimension zero |