Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $400$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot10^{2}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20I1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.72.1.47 |
Level structure
| $\GL_2(\Z/20\Z)$-generators: | $\begin{bmatrix}13&0\\2&7\end{bmatrix}$, $\begin{bmatrix}13&5\\2&7\end{bmatrix}$, $\begin{bmatrix}19&6\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: | $(C_2\times D_{10}):C_4^2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 20-isogeny field degree: | $2$ |
| Cyclic 20-torsion field degree: | $8$ |
| Full 20-torsion field degree: | $640$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 400.2.a.e |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 50x - 125 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
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{60x^{2}y^{22}+1222500x^{2}y^{20}z^{2}+1376562500x^{2}y^{18}z^{4}-25782539062500x^{2}y^{16}z^{6}-10926416015625000x^{2}y^{14}z^{8}+152539625244140625000x^{2}y^{12}z^{10}-15631425018310546875000x^{2}y^{10}z^{12}-189363061237335205078125000x^{2}y^{8}z^{14}-48748323537111282348632812500x^{2}y^{6}z^{16}+3207177407294511795043945312500x^{2}y^{4}z^{18}-63876650612801313400268554687500x^{2}y^{2}z^{20}+414939641486853361129760742187500x^{2}z^{22}+2100xy^{22}z+21987500xy^{20}z^{3}-25976562500xy^{18}z^{5}-388275585937500xy^{16}z^{7}+314495361328125000xy^{14}z^{9}+1718576580810546875000xy^{12}z^{11}-991215790557861328125000xy^{10}z^{13}-1890895965099334716796875000xy^{8}z^{15}-371967231792211532592773437500xy^{6}z^{17}+25506940182298421859741210937500xy^{4}z^{19}-513937751669436693191528320312500xy^{2}z^{21}+3356932208407670259475708007812500xz^{23}+y^{24}+54500y^{22}z^{2}+239656250y^{20}z^{4}-1248710937500y^{18}z^{6}-3167240478515625y^{16}z^{8}+10098899658203125000y^{14}z^{10}+7420886398315429687500y^{12}z^{12}-16866209712982177734375000y^{10}z^{14}-8576609529554843902587890625y^{8}z^{16}-393547005385160446166992187500y^{6}z^{18}+42458977947011590003967285156250y^{4}z^{20}-941073927562683820724487304687500y^{2}z^{22}+6411170019418932497501373291015625z^{24}}{z^{3}y^{2}(2000x^{2}y^{16}z+17525000x^{2}y^{14}z^{3}+47453125000x^{2}y^{12}z^{5}+60770703125000x^{2}y^{10}z^{7}+42989697265625000x^{2}y^{8}z^{9}+17810638427734375000x^{2}y^{6}z^{11}+4310201263427734375000x^{2}y^{4}z^{13}+564874172210693359375000x^{2}y^{2}z^{15}+30995905399322509765625000x^{2}z^{17}+xy^{18}+50625xy^{16}z^{2}+288687500xy^{14}z^{4}+626992187500xy^{12}z^{6}+695751464843750xy^{10}z^{8}+443641540527343750xy^{8}z^{10}+169644699096679687500xy^{6}z^{12}+38489957809448242187500xy^{4}z^{14}+4781717360019683837890625xy^{2}z^{16}+250762142241001129150390625xz^{18}+60y^{18}z+978125y^{16}z^{3}+3550625000y^{14}z^{5}+5597773437500y^{12}z^{7}+4746230468750000y^{10}z^{9}+2364906921386718750y^{8}z^{11}+711748504638671875000y^{6}z^{13}+126611318588256835937500y^{4}z^{15}+12154608964920043945312500y^{2}z^{17}+478913076221942901611328125z^{19})}$ |
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 |
|---|---|---|---|---|---|---|
| 20.36.0.b.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.36.0.d.1 | $20$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 20.36.1.h.1 | $20$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 20.144.5.g.1 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.144.5.m.2 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.144.5.r.1 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.144.5.w.1 | $20$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 20.360.17.a.1 | $20$ | $5$ | $5$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 40.144.5.by.1 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 40.144.5.dm.2 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.144.5.eu.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.144.5.ge.1 | $40$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 60.144.5.ng.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.nk.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.no.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.144.5.ns.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.216.9.ct.2 | $60$ | $3$ | $3$ | $9$ | $3$ | $1^{4}\cdot2^{2}$ |
| 60.288.17.kg.2 | $60$ | $4$ | $4$ | $17$ | $4$ | $1^{8}\cdot2^{4}$ |
| 100.360.17.c.1 | $100$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.144.5.dvo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dwq.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dxs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dyu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.ho.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.hp.2 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.hw.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.144.5.hx.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.144.5.ho.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.144.5.hp.2 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.144.5.hw.2 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.144.5.hx.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.144.5.ho.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.144.5.hp.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.144.5.hw.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.144.5.hx.2 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.ces.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.cez.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.cgw.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.144.5.chd.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |