Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $5760$ | $\PSL_2$-index: | $2880$ | ||||
| Genus: | $169 = 1 + \frac{ 2880 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 144 }{2}$ | ||||||
| Cusps: | $144$ (of which $8$ are rational) | Cusp widths | $20^{144}$ | Cusp orbits | $1^{8}\cdot2^{8}\cdot4^{12}\cdot8^{9}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $29 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $29 \le \gamma \le 48$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.5760.169.25 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}1&0\\0&11\end{bmatrix}$, $\begin{bmatrix}1&0\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $C_2\times C_4$ |
| Contains $-I$: | no $\quad$ (see 20.2880.169.c.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $1$ |
| Cyclic 20-torsion field degree: | $1$ |
| Full 20-torsion field degree: | $8$ |
Jacobian
| Conductor: | $2^{460}\cdot5^{288}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{43}\cdot2^{39}\cdot4^{4}\cdot8^{4}$ |
| Newforms: | 20.2.a.a$^{6}$, 20.2.e.a$^{6}$, 40.2.a.a$^{4}$, 40.2.c.a$^{4}$, 50.2.a.a$^{4}$, 50.2.a.b$^{4}$, 50.2.b.a$^{4}$, 80.2.a.a$^{2}$, 80.2.a.b$^{2}$, 80.2.c.a$^{2}$, 80.2.n.a$^{2}$, 80.2.n.b$^{2}$, 100.2.a.a$^{3}$, 100.2.c.a$^{3}$, 100.2.e.a$^{3}$, 100.2.e.b$^{3}$, 100.2.e.c$^{3}$, 100.2.e.d$^{3}$, 200.2.a.a$^{2}$, 200.2.a.b$^{2}$, 200.2.a.c$^{2}$, 200.2.a.d$^{2}$, 200.2.a.e$^{2}$, 200.2.c.a$^{2}$, 200.2.c.b$^{2}$, 400.2.a.a, 400.2.a.b, 400.2.a.c, 400.2.a.d, 400.2.a.e, 400.2.a.f, 400.2.a.g, 400.2.a.h, 400.2.c.a, 400.2.c.b, 400.2.c.c, 400.2.c.d, 400.2.n.a, 400.2.n.b, 400.2.n.c, 400.2.n.d |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(4)$ | $4$ | $120$ | $120$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{arith}}(5)$ | $5$ | $48$ | $48$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.1152.25-20.c.1.4 | $20$ | $5$ | $5$ | $25$ | $0$ | $1^{36}\cdot2^{32}\cdot4^{3}\cdot8^{4}$ |
| 20.1152.25-20.c.2.4 | $20$ | $5$ | $5$ | $25$ | $0$ | $1^{36}\cdot2^{32}\cdot4^{3}\cdot8^{4}$ |
| 20.2880.73-20.a.1.3 | $20$ | $2$ | $2$ | $73$ | $3$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.73-20.a.1.8 | $20$ | $2$ | $2$ | $73$ | $3$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.73-20.b.1.3 | $20$ | $2$ | $2$ | $73$ | $1$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.73-20.b.1.6 | $20$ | $2$ | $2$ | $73$ | $1$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.73-20.b.2.4 | $20$ | $2$ | $2$ | $73$ | $1$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.73-20.b.2.5 | $20$ | $2$ | $2$ | $73$ | $1$ | $1^{24}\cdot2^{20}\cdot4^{4}\cdot8^{2}$ |
| 20.2880.85-20.e.1.1 | $20$ | $2$ | $2$ | $85$ | $5$ | $2^{18}\cdot4^{4}\cdot8^{4}$ |
| 20.2880.85-20.e.1.6 | $20$ | $2$ | $2$ | $85$ | $5$ | $2^{18}\cdot4^{4}\cdot8^{4}$ |
| 20.2880.85-20.m.1.5 | $20$ | $2$ | $2$ | $85$ | $3$ | $1^{24}\cdot2^{22}\cdot8^{2}$ |
| 20.2880.85-20.m.1.8 | $20$ | $2$ | $2$ | $85$ | $3$ | $1^{24}\cdot2^{22}\cdot8^{2}$ |
| 20.2880.85-20.n.1.1 | $20$ | $2$ | $2$ | $85$ | $1$ | $1^{24}\cdot2^{22}\cdot8^{2}$ |
| 20.2880.85-20.n.1.3 | $20$ | $2$ | $2$ | $85$ | $1$ | $1^{24}\cdot2^{22}\cdot8^{2}$ |
| 20.2880.85-20.n.1.8 | $20$ | $2$ | $2$ | $85$ | $1$ | $1^{24}\cdot2^{22}\cdot8^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.11520.361-40.c.1.2 | $40$ | $2$ | $2$ | $361$ | $9$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.e.1.2 | $40$ | $2$ | $2$ | $361$ | $13$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.m.1.4 | $40$ | $2$ | $2$ | $361$ | $13$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.s.1.1 | $40$ | $2$ | $2$ | $361$ | $9$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.y.1.12 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.y.2.9 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.y.3.14 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.y.4.9 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.ba.1.11 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.ba.2.11 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.ba.3.13 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.ba.4.11 | $40$ | $2$ | $2$ | $361$ | $11$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.bw.1.11 | $40$ | $2$ | $2$ | $361$ | $13$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.cc.1.11 | $40$ | $2$ | $2$ | $361$ | $9$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.ck.1.9 | $40$ | $2$ | $2$ | $361$ | $9$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.361-40.cm.1.12 | $40$ | $2$ | $2$ | $361$ | $13$ | $2^{20}\cdot4^{22}\cdot8^{8}$ |
| 40.11520.385-40.ib.1.11 | $40$ | $2$ | $2$ | $385$ | $35$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ib.2.14 | $40$ | $2$ | $2$ | $385$ | $35$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ib.3.9 | $40$ | $2$ | $2$ | $385$ | $35$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ic.1.11 | $40$ | $2$ | $2$ | $385$ | $24$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ic.2.14 | $40$ | $2$ | $2$ | $385$ | $24$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ic.3.9 | $40$ | $2$ | $2$ | $385$ | $24$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ij.1.11 | $40$ | $2$ | $2$ | $385$ | $25$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ij.2.14 | $40$ | $2$ | $2$ | $385$ | $25$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ij.3.9 | $40$ | $2$ | $2$ | $385$ | $25$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ik.1.11 | $40$ | $2$ | $2$ | $385$ | $39$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ik.2.14 | $40$ | $2$ | $2$ | $385$ | $39$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.385-40.ik.3.9 | $40$ | $2$ | $2$ | $385$ | $39$ | $1^{40}\cdot2^{56}\cdot4^{12}\cdot8^{2}$ |
| 40.11520.409-40.hc.1.21 | $40$ | $2$ | $2$ | $409$ | $11$ | $2^{18}\cdot4^{21}\cdot8^{15}$ |
| 40.11520.409-40.hc.2.19 | $40$ | $2$ | $2$ | $409$ | $11$ | $2^{18}\cdot4^{21}\cdot8^{15}$ |
| 40.11520.409-40.hz.1.21 | $40$ | $2$ | $2$ | $409$ | $9$ | $2^{18}\cdot4^{21}\cdot8^{15}$ |
| 40.11520.409-40.hz.2.19 | $40$ | $2$ | $2$ | $409$ | $9$ | $2^{18}\cdot4^{21}\cdot8^{15}$ |
| 60.17280.649-60.fh.1.9 | $60$ | $3$ | $3$ | $649$ | $51$ | $1^{108}\cdot2^{90}\cdot8^{24}$ |
| 60.23040.817-60.bb.1.13 | $60$ | $4$ | $4$ | $817$ | $30$ | $1^{162}\cdot2^{99}\cdot4^{7}\cdot8^{19}\cdot12^{9}$ |