Invariants
| Level: | $20$ | $\SL_2$-level: | $10$ | Newform level: | $400$ | ||
| 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 $2$ are rational) | Cusp widths | $2^{6}\cdot10^{6}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| 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: | 10K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.144.1.25 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}1&10\\8&7\end{bmatrix}$, $\begin{bmatrix}5&7\\12&3\end{bmatrix}$, $\begin{bmatrix}9&5\\10&9\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $(C_2\times D_{20}):C_4$ |
| Contains $-I$: | no $\quad$ (see 20.72.1.i.1 for the level structure with $-I$) |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $8$ |
| Full 20-torsion field degree: | $320$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 33x - 62 $ |
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 |
|---|
| $(0:1:0)$, $(-2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5^5}\cdot\frac{120x^{2}y^{22}+4435000x^{2}y^{20}z^{2}+4140625000x^{2}y^{18}z^{4}-23216484375000x^{2}y^{16}z^{6}+4729199218750000x^{2}y^{14}z^{8}+16002644042968750000x^{2}y^{12}z^{10}-3728815612792968750000x^{2}y^{10}z^{12}-2776815605163574218750000x^{2}y^{8}z^{14}-237619783878326416015625000x^{2}y^{6}z^{16}+6099186837673187255859375000x^{2}y^{4}z^{18}-46688430011272430419921875000x^{2}y^{2}z^{20}+116196461021900177001953125000x^{2}z^{22}+6480xy^{22}z+76390000xy^{20}z^{3}-34281250000xy^{18}z^{5}-244869843750000xy^{16}z^{7}+163025195312500000xy^{14}z^{9}+112093217773437500000xy^{12}z^{11}-49928516235351562500000xy^{10}z^{13}-22490076828002929687500000xy^{8}z^{15}-1615977988243103027343750000xy^{6}z^{17}+42724773287773132324218750000xy^{4}z^{19}-329743400216102600097656250000xy^{2}z^{21}+823857262730598449707031250000xz^{23}+y^{24}+203980y^{22}z^{2}+777571250y^{20}z^{4}-1581101562500y^{18}z^{6}-863471650390625y^{16}z^{8}+1912211474609375000y^{14}z^{10}+10419862670898437500y^{12}z^{12}-384461060943603515625000y^{10}z^{14}-64299306738376617431640625y^{8}z^{16}-1519628301858901977539062500y^{6}z^{18}+55262239649891853332519531250y^{4}z^{20}-458368826657533645629882812500y^{2}z^{22}+1182943233288824558258056640625z^{24}}{z^{3}y^{2}(y^{2}+125z^{2})^{5}(3250x^{2}y^{6}z+13168750x^{2}y^{4}z^{3}+6652343750x^{2}y^{2}z^{5}+744628906250x^{2}z^{7}+xy^{8}+75500xy^{6}z^{2}+147518750xy^{4}z^{4}+55398437500xy^{2}z^{6}+5279541015625xz^{8}+87y^{8}z+1109875y^{6}z^{3}+938846875y^{4}z^{5}+176228515625y^{2}z^{7}+7580566406250z^{9})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 10.72.0-10.a.2.4 | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.72.0-10.a.2.7 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.48.1-20.e.1.2 | $20$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.288.5-20.s.1.3 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.288.5-20.w.1.5 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.288.5-20.ba.1.3 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.288.5-20.be.1.5 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 20.720.13-20.s.1.1 | $20$ | $5$ | $5$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 40.288.5-40.ez.1.7 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 40.288.5-40.gb.1.7 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 40.288.5-40.ig.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.288.5-40.ji.1.7 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.ie.1.7 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.288.5-60.im.1.5 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.288.5-60.oo.1.7 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.288.5-60.ow.1.5 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.432.13-60.fo.2.11 | $60$ | $3$ | $3$ | $13$ | $0$ | $1^{6}\cdot2^{3}$ |
| 60.576.13-60.ne.2.13 | $60$ | $4$ | $4$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 120.288.5-120.cmp.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cot.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.edk.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.efo.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.ds.1.7 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.du.1.5 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.ey.1.7 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.288.5-140.fa.1.5 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.ds.1.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.du.1.3 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.ey.1.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.288.5-220.fa.1.3 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.ds.1.3 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.du.1.5 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.ey.1.3 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.288.5-260.fa.1.5 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bdp.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bed.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bme.1.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.288.5-280.bms.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |