Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | Newform level: | $20$ | ||
| Index: | $144$ | $\PSL_2$-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 | $2^{6}\cdot10^{6}$ | 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: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.144.1.1 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}1&4\\0&9\end{bmatrix}$, $\begin{bmatrix}1&8\\0&3\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $F_5$ |
| Contains $-I$: | no $\quad$ (see 10.72.1.a.1 for the level structure with $-I$) |
| Cyclic 10-isogeny field degree: | $1$ |
| Cyclic 10-torsion field degree: | $1$ |
| Full 10-torsion field degree: | $20$ |
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 $ |
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 |
|---|
| $(1:1:1)$, $(-1:1:1)$, $(1:-1:1)$, $(-1:-1:1)$, $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{24x^{2}y^{22}-7096x^{2}y^{20}z^{2}+53000x^{2}y^{18}z^{4}+2377368x^{2}y^{16}z^{6}+3874160x^{2}y^{14}z^{8}-104874928x^{2}y^{12}z^{10}-195497328x^{2}y^{10}z^{12}+1164680880x^{2}y^{8}z^{14}-797319688x^{2}y^{6}z^{16}-163723800x^{2}y^{4}z^{18}-10026264x^{2}y^{2}z^{20}-199624x^{2}z^{22}-240xy^{22}z+18768xy^{20}z^{3}+130160xy^{18}z^{5}-3113040xy^{16}z^{7}-23610720xy^{14}z^{9}+63022880xy^{12}z^{11}+367140576xy^{10}z^{13}-954859680xy^{8}z^{15}+446608720xy^{6}z^{17}+98397840xy^{4}z^{19}+6141360xy^{2}z^{21}+123376xz^{23}-y^{24}+1532y^{22}z^{2}-41122y^{20}z^{4}-765940y^{18}z^{6}+1911185y^{16}z^{8}+52595192y^{14}z^{10}+39257636y^{12}z^{12}-628137480y^{10}z^{14}+510470545y^{8}z^{16}+102253580y^{6}z^{18}+6217566y^{4}z^{20}+123388y^{2}z^{22}-z^{24}}{z^{3}y^{2}(y-z)^{5}(y+z)^{5}(130x^{2}y^{6}z-4214x^{2}y^{4}z^{3}+17030x^{2}y^{2}z^{5}-15250x^{2}z^{7}+xy^{8}-500xy^{6}z^{2}+6070xy^{4}z^{4}-14740xy^{2}z^{6}+9425xz^{8}-17y^{8}z+1555y^{6}z^{3}-8915y^{4}z^{5}+9425y^{2}z^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| $X_1(5)$ | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.48.1-10.a.1.2 | $10$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| $X_1(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.72.0-10.a.2.4 | $10$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(10)$ | $10$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 20.288.3-20.a.1.2 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.288.3-20.b.2.1 | $20$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 20.288.5-20.a.1.4 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.288.5-20.b.1.4 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.288.5-20.e.1.4 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.288.5-20.f.1.3 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.288.7-20.f.2.8 | $20$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 20.288.7-20.g.1.8 | $20$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 30.432.13-30.a.2.2 | $30$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 30.576.13-30.a.1.1 | $30$ | $4$ | $4$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 40.288.3-40.a.2.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.3-40.b.1.4 | $40$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 40.288.5-40.a.1.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.d.1.4 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.m.1.4 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.p.1.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.7-40.k.2.15 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 40.288.7-40.l.1.15 | $40$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 50.720.13-50.a.1.2 | $50$ | $5$ | $5$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.288.3-60.b.2.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.288.3-60.c.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 60.288.5-60.y.1.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.288.5-60.z.1.4 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.bk.1.4 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.bl.1.4 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.7-60.eh.1.15 | $60$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 60.288.7-60.ei.2.15 | $60$ | $2$ | $2$ | $7$ | $0$ | $2\cdot4$ |
| 70.1152.37-70.a.2.2 | $70$ | $8$ | $8$ | $37$ | $0$ | $1^{12}\cdot2^{8}\cdot4^{2}$ |
| 70.3024.109-70.a.1.4 | $70$ | $21$ | $21$ | $109$ | $14$ | $1^{12}\cdot2^{28}\cdot4^{10}$ |
| 70.4032.145-70.a.1.2 | $70$ | $28$ | $28$ | $145$ | $14$ | $1^{24}\cdot2^{36}\cdot4^{12}$ |
| 120.288.3-120.b.2.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.3-120.c.1.4 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.288.5-120.cu.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cx.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ee.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eh.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.7-120.yh.2.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.yi.1.25 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.288.3-140.a.2.2 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.288.3-140.b.1.3 | $140$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 140.288.5-140.a.1.4 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.b.1.4 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.e.1.4 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.f.1.4 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.7-140.a.1.15 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.288.7-140.b.2.15 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.288.3-220.a.1.8 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.288.3-220.b.1.8 | $220$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 220.288.5-220.a.1.12 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.b.1.12 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.e.1.12 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.f.1.12 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.7-220.a.1.14 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.288.7-220.b.1.16 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.288.3-260.a.1.3 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.288.3-260.b.2.2 | $260$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 260.288.5-260.a.1.4 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.b.1.4 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.e.1.4 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.f.1.4 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.7-260.a.2.15 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.288.7-260.b.1.15 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.3-280.a.2.2 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.3-280.b.1.3 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.288.5-280.a.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.d.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.m.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.p.1.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.7-280.a.2.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.288.7-280.b.1.25 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |