Invariants
Level: | $10$ | $\SL_2$-level: | $10$ | Newform level: | $100$ | ||
Index: | $90$ | $\PSL_2$-index: | $90$ | ||||
Genus: | $3 = 1 + \frac{ 90 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 9 }{2}$ | ||||||
Cusps: | $9$ (of which $1$ is rational) | Cusp widths | $10^{9}$ | Cusp orbits | $1\cdot2^{2}\cdot4$ | ||
Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $3$ | ||||||
$\overline{\Q}$-gonality: | $3$ | ||||||
Rational cusps: | $1$ | ||||||
Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
Cummins and Pauli (CP) label: | 10C3 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.90.3.2 |
Level structure
$\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}1&7\\0&7\end{bmatrix}$, $\begin{bmatrix}3&5\\4&7\end{bmatrix}$ |
$\GL_2(\Z/10\Z)$-subgroup: | $C_4\wr C_2$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 10-isogeny field degree: | $2$ |
Cyclic 10-torsion field degree: | $8$ |
Full 10-torsion field degree: | $32$ |
Jacobian
Conductor: | $2^{4}\cdot5^{6}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{3}$ |
Newforms: | 50.2.a.b$^{2}$, 100.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
$ 0 $ | $=$ | $ 5 x^{4} + 5 x^{3} y - x^{3} z - 5 x^{2} y^{2} + x^{2} y z + 2 x^{2} z^{2} + 3 x y^{2} z + 2 x y z^{2} + \cdots + 2 y z^{3} $ |
Rational points
This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Elliptic curve | CM | $j$-invariant | $j$-height | Canonical model | |
---|---|---|---|---|---|
no | $\infty$ | $0.000$ | $(0:0:1)$ | ||
32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 90 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{113975249760055541992187500x^{3}y^{20}-70440589193248748779296875x^{2}y^{21}+7580566406250000000xy^{22}-659179687500000000y^{23}-283518025070285797119140625x^{3}y^{19}z+152428490433979034423828125x^{2}y^{20}z+59678375813961029052734375xy^{21}z-28176239434623718261718750y^{22}z+304741883107906341552734375x^{3}y^{18}z^{2}-131638748697818756103515625x^{2}y^{19}z^{2}-102861077497554779052734375xy^{20}z^{2}+41913367391891479492187500y^{21}z^{2}-293895927426850891113281250x^{3}y^{17}z^{3}+134777869531970977783203125x^{2}y^{18}z^{3}+14652806332236480712890625xy^{19}z^{3}+22931037741476440429687500y^{20}z^{3}+349266166732661437988281250x^{3}y^{16}z^{4}-195993939145408325195312500x^{2}y^{17}z^{4}+52237572679567260742187500xy^{18}z^{4}-74916934964787597656250000y^{19}z^{4}-325981556652968139648437500x^{3}y^{15}z^{5}+177917921087999877929687500x^{2}y^{16}z^{5}-30389779863496398925781250xy^{17}z^{5}+65444992841897888183593750y^{18}z^{5}+220528289592103118896484375x^{3}y^{14}z^{6}-111808663184489959716796875x^{2}y^{15}z^{6}+52577168499277587890625000xy^{16}z^{6}-77396314809919067382812500y^{17}z^{6}-139920496128335333251953125x^{3}y^{13}z^{7}+82406542758201287841796875x^{2}y^{14}z^{7}-101684497185335198974609375xy^{15}z^{7}+105420044439286621093750000y^{16}z^{7}+90090094746084793701171875x^{3}y^{12}z^{8}-63773547034043560791015625x^{2}y^{13}z^{8}+94452551516520158691406250xy^{14}z^{8}-93014724445851809082031250y^{15}z^{8}-47578997664911669921875000x^{3}y^{11}z^{9}+34899260737571136962890625x^{2}y^{12}z^{9}-60016908328414666259765625xy^{13}z^{9}+60049430163791393554687500y^{14}z^{9}+19297421425820170068359375x^{3}y^{10}z^{10}-14369001687137275537109375x^{2}y^{11}z^{10}+37806090359136194140625000xy^{12}z^{10}-37647021844402229687500000y^{13}z^{10}-7077711006754608935546875x^{3}y^{9}z^{11}+6359936397507185888671875x^{2}y^{10}z^{11}-23040065520597134326171875xy^{11}z^{11}+22790509104169185156250000y^{12}z^{11}+2623317203536420511328125x^{3}y^{8}z^{12}-2906966300575551422265625x^{2}y^{9}z^{12}+10869111034225395006250000xy^{10}z^{12}-10953591144144702089843750y^{11}z^{12}-824340888599372836875000x^{3}y^{7}z^{13}+952395303123496881640625x^{2}y^{8}z^{13}-3914360914780374534453125xy^{9}z^{13}+4082139645389090486562500y^{10}z^{13}+188099924106218311015625x^{3}y^{6}z^{14}-210068097686610531453125x^{2}y^{7}z^{14}+1285399034266817602375000xy^{8}z^{14}-1383108537039170005187500y^{9}z^{14}-35132479147493217971875x^{3}y^{5}z^{15}+49278150001112303628125x^{2}y^{6}z^{15}-422690933960485723865625xy^{7}z^{15}+465786220506943711525000y^{8}z^{15}+7428064372447141940625x^{3}y^{4}z^{16}-15520862674213323212500x^{2}y^{5}z^{16}+116288245952654951156250xy^{6}z^{16}-132232696450872305018750y^{7}z^{16}-1479542197124669155875x^{3}y^{3}z^{17}+3299930372214281107250x^{2}y^{4}z^{17}-22820504796585274074750xy^{5}z^{17}+27147406124241664771250y^{6}z^{17}+167114291111295296650x^{3}y^{2}z^{18}-251961550474790421000x^{2}y^{3}z^{18}+3484821623476465495075xy^{4}z^{18}-4390024853962097526100y^{5}z^{18}-5133157705629185275x^{3}yz^{19}-17624868553315320480x^{2}y^{2}z^{19}-577287665314965547430xy^{3}z^{19}+778612113263586364580y^{4}z^{19}-308225143707089311x^{3}z^{20}+2068447745746849301x^{2}yz^{20}+92425678884820064448xy^{2}z^{20}-135822389450107423162y^{3}z^{20}+132723627037108787x^{2}z^{21}-8445065094572693033xyz^{21}+13928760909859031878y^{2}z^{21}+254873038637229404xz^{22}-509746077274458808yz^{22}+762939453125000z^{23}}{-4005432128906250x^{3}y^{20}-3623962402343750x^{2}y^{21}+4386901855468750xy^{22}-381469726562500y^{23}+56076049804687500x^{3}y^{19}z+45394897460937500x^{2}y^{20}z-66757202148437500xy^{21}z+10681152343750000y^{22}z-248096466064453125x^{3}y^{18}z^{2}-174129486083984375x^{2}y^{19}z^{2}+322063446044921875xy^{20}z^{2}-73966979980468750y^{21}z^{2}+148926544189453125x^{3}y^{17}z^{3}+22747039794921875x^{2}y^{18}z^{3}-276631927490234375xy^{19}z^{3}+125607299804687500y^{20}z^{3}+1904108428955078125x^{3}y^{16}z^{4}+1400445404052734375x^{2}y^{17}z^{4}-2386752471923828125xy^{18}z^{4}+509118041992187500y^{19}z^{4}-4846704711914062500x^{3}y^{15}z^{5}-2723778228759765625x^{2}y^{16}z^{5}+6876526641845703125xy^{17}z^{5}-2134574279785156250y^{18}z^{5}-1575913391113281250x^{3}y^{14}z^{6}-2144330261230468750x^{2}y^{15}z^{6}+1070019561767578125xy^{16}z^{6}+534325378417968750y^{17}z^{6}+21213220446777343750x^{3}y^{13}z^{7}+12040379382324218750x^{2}y^{14}z^{7}-29691545397949218750xy^{15}z^{7}+9317892395019531250y^{16}z^{7}-21513688481445312500x^{3}y^{12}z^{8}-7808592218017578125x^{2}y^{13}z^{8}+33347121689453125000xy^{14}z^{8}-13668154155273437500y^{15}z^{8}-29109447515869140625x^{3}y^{11}z^{9}-17237565207763671875x^{2}y^{12}z^{9}+40656787038330078125xy^{13}z^{9}-12567659102539062500y^{14}z^{9}+371483554017871093750x^{3}y^{10}z^{10}-154467463371630859375x^{2}y^{11}z^{10}-113123343830468750000xy^{12}z^{10}+46435334519628906250y^{13}z^{10}-762031905909277343750x^{3}y^{9}z^{11}+394067137230908203125x^{2}y^{10}z^{11}+192100634868798828125xy^{11}z^{11}-93012894428515625000y^{12}z^{11}+707451048367198437500x^{3}y^{8}z^{12}-369896863405184375000x^{2}y^{9}z^{12}-133720503915753125000xy^{10}z^{12}+48912943211445312500y^{11}z^{12}-364967003731801093750x^{3}y^{7}z^{13}+195931142096724218750x^{2}y^{8}z^{13}-127932163370142343750xy^{9}z^{13}+142094764962002031250y^{10}z^{13}+117496610321254062500x^{3}y^{6}z^{14}-86923042516790562500x^{2}y^{7}z^{14}+288013940098923968750xy^{8}z^{14}-265863563017853812500y^{9}z^{14}-27629451819866850000x^{3}y^{5}z^{15}+40868545599569100000x^{2}y^{6}z^{15}-213208539141497512500xy^{7}z^{15}+202960449062742018750y^{8}z^{15}+5671331000019631250x^{3}y^{4}z^{16}-14937621480440940625x^{2}y^{5}z^{16}+83257116885804968750xy^{6}z^{16}-85033216833413600000y^{7}z^{16}-938565111289243625x^{3}y^{3}z^{17}+3063979612170620125x^{2}y^{4}z^{17}-18737908084357473875xy^{5}z^{17}+21065963554734392500y^{6}z^{17}+89260739638374300x^{3}y^{2}z^{18}-242078331888950750x^{2}y^{3}z^{18}+2562395037167538150xy^{4}z^{18}-3288495117910474950y^{5}z^{18}-2759450022520925x^{3}yz^{19}-11013546160349410x^{2}y^{2}z^{19}-258096938827527560xy^{3}z^{19}+392518515350571610y^{4}z^{19}-50526243787787x^{3}z^{20}+2157720416267617x^{2}yz^{20}+25620907563385991xy^{2}z^{20}-44087608390324004y^{3}z^{20}-36358703194121x^{2}z^{21}-1824455389448811xyz^{21}+3362543234570426y^{2}z^{21}+43180235710368xz^{22}-86360471420736yz^{22}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
10.30.1.b.1 | $10$ | $3$ | $3$ | $1$ | $0$ | $1^{2}$ |
10.45.1.a.1 | $10$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
10.180.7.b.1 | $10$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
10.180.7.c.1 | $10$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
20.180.7.e.1 | $20$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
20.180.7.q.1 | $20$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
20.180.8.a.1 | $20$ | $2$ | $2$ | $8$ | $0$ | $1^{5}$ |
20.180.8.d.1 | $20$ | $2$ | $2$ | $8$ | $2$ | $1^{5}$ |
20.180.8.j.1 | $20$ | $2$ | $2$ | $8$ | $0$ | $1^{5}$ |
20.180.8.l.1 | $20$ | $2$ | $2$ | $8$ | $2$ | $1^{5}$ |
20.180.10.v.1 | $20$ | $2$ | $2$ | $10$ | $2$ | $1^{7}$ |
20.180.10.x.1 | $20$ | $2$ | $2$ | $10$ | $1$ | $1^{7}$ |
20.180.10.bi.1 | $20$ | $2$ | $2$ | $10$ | $1$ | $1^{7}$ |
20.180.10.bl.1 | $20$ | $2$ | $2$ | $10$ | $1$ | $1^{7}$ |
30.180.7.e.1 | $30$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
30.180.7.i.1 | $30$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
30.270.16.j.1 | $30$ | $3$ | $3$ | $16$ | $5$ | $1^{11}\cdot2$ |
30.360.22.g.1 | $30$ | $4$ | $4$ | $22$ | $1$ | $1^{19}$ |
40.180.7.j.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
40.180.7.u.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
40.180.7.by.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
40.180.7.ch.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
40.180.8.a.1 | $40$ | $2$ | $2$ | $8$ | $3$ | $1^{5}$ |
40.180.8.j.1 | $40$ | $2$ | $2$ | $8$ | $2$ | $1^{5}$ |
40.180.8.bc.1 | $40$ | $2$ | $2$ | $8$ | $0$ | $1^{5}$ |
40.180.8.bi.1 | $40$ | $2$ | $2$ | $8$ | $5$ | $1^{5}$ |
40.180.10.dd.1 | $40$ | $2$ | $2$ | $10$ | $4$ | $1^{7}$ |
40.180.10.dj.1 | $40$ | $2$ | $2$ | $10$ | $3$ | $1^{7}$ |
40.180.10.em.1 | $40$ | $2$ | $2$ | $10$ | $2$ | $1^{7}$ |
40.180.10.ev.1 | $40$ | $2$ | $2$ | $10$ | $5$ | $1^{7}$ |
50.450.27.a.1 | $50$ | $5$ | $5$ | $27$ | $12$ | $2^{5}\cdot4^{2}\cdot6$ |
50.2250.161.a.1 | $50$ | $25$ | $25$ | $161$ | $72$ | $1^{6}\cdot2^{28}\cdot4^{12}\cdot8^{6}$ |
60.180.7.be.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
60.180.7.cg.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
60.180.8.a.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{5}$ |
60.180.8.d.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{5}$ |
60.180.8.be.1 | $60$ | $2$ | $2$ | $8$ | $2$ | $1^{5}$ |
60.180.8.bg.1 | $60$ | $2$ | $2$ | $8$ | $2$ | $1^{5}$ |
60.180.10.ck.1 | $60$ | $2$ | $2$ | $10$ | $4$ | $1^{7}$ |
60.180.10.cm.1 | $60$ | $2$ | $2$ | $10$ | $2$ | $1^{7}$ |
60.180.10.ed.1 | $60$ | $2$ | $2$ | $10$ | $4$ | $1^{7}$ |
60.180.10.eg.1 | $60$ | $2$ | $2$ | $10$ | $3$ | $1^{7}$ |
70.180.7.m.1 | $70$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
70.180.7.n.1 | $70$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
70.720.52.bl.1 | $70$ | $8$ | $8$ | $52$ | $10$ | $1^{29}\cdot2^{10}$ |
70.1890.140.c.1 | $70$ | $21$ | $21$ | $140$ | $68$ | $1^{25}\cdot2^{32}\cdot3^{4}\cdot4^{9}$ |
70.2520.189.k.1 | $70$ | $28$ | $28$ | $189$ | $78$ | $1^{54}\cdot2^{42}\cdot3^{4}\cdot4^{9}$ |
110.180.7.i.1 | $110$ | $2$ | $2$ | $7$ | $?$ | not computed |
110.180.7.j.1 | $110$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.180.7.dx.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.180.7.ed.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.180.7.hm.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.180.7.hv.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.180.8.a.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.180.8.j.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.180.8.dm.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.180.8.ds.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.180.10.hv.1 | $120$ | $2$ | $2$ | $10$ | $?$ | not computed |
120.180.10.ib.1 | $120$ | $2$ | $2$ | $10$ | $?$ | not computed |
120.180.10.mw.1 | $120$ | $2$ | $2$ | $10$ | $?$ | not computed |
120.180.10.nf.1 | $120$ | $2$ | $2$ | $10$ | $?$ | not computed |
130.180.7.m.1 | $130$ | $2$ | $2$ | $7$ | $?$ | not computed |
130.180.7.n.1 | $130$ | $2$ | $2$ | $7$ | $?$ | not computed |
140.180.7.ck.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
140.180.7.cr.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
140.180.8.f.1 | $140$ | $2$ | $2$ | $8$ | $?$ | not computed |
140.180.8.h.1 | $140$ | $2$ | $2$ | $8$ | $?$ | not computed |
140.180.8.n.1 | $140$ | $2$ | $2$ | $8$ | $?$ | not computed |
140.180.8.p.1 | $140$ | $2$ | $2$ | $8$ | $?$ | not computed |
140.180.10.eb.1 | $140$ | $2$ | $2$ | $10$ | $?$ | not computed |
140.180.10.ed.1 | $140$ | $2$ | $2$ | $10$ | $?$ | not computed |
140.180.10.ej.1 | $140$ | $2$ | $2$ | $10$ | $?$ | not computed |
140.180.10.el.1 | $140$ | $2$ | $2$ | $10$ | $?$ | not computed |
170.180.7.i.1 | $170$ | $2$ | $2$ | $7$ | $?$ | not computed |
170.180.7.j.1 | $170$ | $2$ | $2$ | $7$ | $?$ | not computed |
190.180.7.m.1 | $190$ | $2$ | $2$ | $7$ | $?$ | not computed |
190.180.7.n.1 | $190$ | $2$ | $2$ | $7$ | $?$ | not computed |
210.180.7.be.1 | $210$ | $2$ | $2$ | $7$ | $?$ | not computed |
210.180.7.bh.1 | $210$ | $2$ | $2$ | $7$ | $?$ | not computed |
220.180.7.cg.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
220.180.7.cn.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
220.180.8.f.1 | $220$ | $2$ | $2$ | $8$ | $?$ | not computed |
220.180.8.h.1 | $220$ | $2$ | $2$ | $8$ | $?$ | not computed |
220.180.8.n.1 | $220$ | $2$ | $2$ | $8$ | $?$ | not computed |
220.180.8.p.1 | $220$ | $2$ | $2$ | $8$ | $?$ | not computed |
220.180.10.eb.1 | $220$ | $2$ | $2$ | $10$ | $?$ | not computed |
220.180.10.ed.1 | $220$ | $2$ | $2$ | $10$ | $?$ | not computed |
220.180.10.ej.1 | $220$ | $2$ | $2$ | $10$ | $?$ | not computed |
220.180.10.el.1 | $220$ | $2$ | $2$ | $10$ | $?$ | not computed |
230.180.7.i.1 | $230$ | $2$ | $2$ | $7$ | $?$ | not computed |
230.180.7.j.1 | $230$ | $2$ | $2$ | $7$ | $?$ | not computed |
260.180.7.ck.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
260.180.7.cr.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
260.180.8.f.1 | $260$ | $2$ | $2$ | $8$ | $?$ | not computed |
260.180.8.h.1 | $260$ | $2$ | $2$ | $8$ | $?$ | not computed |
260.180.8.n.1 | $260$ | $2$ | $2$ | $8$ | $?$ | not computed |
260.180.8.p.1 | $260$ | $2$ | $2$ | $8$ | $?$ | not computed |
260.180.10.eb.1 | $260$ | $2$ | $2$ | $10$ | $?$ | not computed |
260.180.10.ed.1 | $260$ | $2$ | $2$ | $10$ | $?$ | not computed |
260.180.10.ej.1 | $260$ | $2$ | $2$ | $10$ | $?$ | not computed |
260.180.10.el.1 | $260$ | $2$ | $2$ | $10$ | $?$ | not computed |
280.180.7.hx.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.180.7.id.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.180.7.iv.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.180.7.jb.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
280.180.8.q.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
280.180.8.w.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
280.180.8.bo.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
280.180.8.bu.1 | $280$ | $2$ | $2$ | $8$ | $?$ | not computed |
280.180.10.mn.1 | $280$ | $2$ | $2$ | $10$ | $?$ | not computed |
280.180.10.mt.1 | $280$ | $2$ | $2$ | $10$ | $?$ | not computed |
280.180.10.nl.1 | $280$ | $2$ | $2$ | $10$ | $?$ | not computed |
280.180.10.nr.1 | $280$ | $2$ | $2$ | $10$ | $?$ | not computed |
290.180.7.i.1 | $290$ | $2$ | $2$ | $7$ | $?$ | not computed |
290.180.7.j.1 | $290$ | $2$ | $2$ | $7$ | $?$ | not computed |
310.180.7.m.1 | $310$ | $2$ | $2$ | $7$ | $?$ | not computed |
310.180.7.n.1 | $310$ | $2$ | $2$ | $7$ | $?$ | not computed |
330.180.7.ba.1 | $330$ | $2$ | $2$ | $7$ | $?$ | not computed |
330.180.7.bd.1 | $330$ | $2$ | $2$ | $7$ | $?$ | not computed |