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Properties

Label 21.128.1-21.a.4.3
Level $21$
Index $128$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $21$ $\SL_2$-level: $21$ Newform level: $21$
Index: $128$ $\PSL_2$-index:$64$
Genus: $1 = 1 + \frac{ 64 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $1^{2}\cdot3^{2}\cdot7^{2}\cdot21^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $4$ 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: 21F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 21.128.1.6

Level structure

$\GL_2(\Z/21\Z)$-generators: $\begin{bmatrix}5&10\\0&4\end{bmatrix}$, $\begin{bmatrix}19&13\\0&1\end{bmatrix}$
$\GL_2(\Z/21\Z)$-subgroup: $C_{21}:C_6^2$
Contains $-I$: no $\quad$ (see 21.64.1.a.4 for the level structure with $-I$)
Cyclic 21-isogeny field degree: $1$
Cyclic 21-torsion field degree: $12$
Full 21-torsion field degree: $756$

Jacobian

Conductor: $3\cdot7$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 21.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} + x y $ $=$ $ x^{3} + x $
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Rational points

This modular curve has rational points, including 2 rational_cusps and 1 known non-cuspidal non-CM point. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve CM $j$-invariant $j$-heightWeierstrass model
no$\infty$ $0.000$$(0:0:1)$, $(0:1:0)$
162.b1 no$\tfrac{-189613868625}{128}$ $= -1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$$25.968$$(1:-2:1)$, $(1:1:1)$

Maps to other modular curves

$j$-invariant map of degree 64 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{762x^{2}y^{36}+1287086881x^{2}y^{35}z+12111180161445x^{2}y^{34}z^{2}+11760084896120585x^{2}y^{33}z^{3}+2729317684267757057x^{2}y^{32}z^{4}+221618705709593133271x^{2}y^{31}z^{5}+7778759299224101588721x^{2}y^{30}z^{6}+131821906660873931928537x^{2}y^{29}z^{7}+1099208774432922625266767x^{2}y^{28}z^{8}+3837204310501841117397712x^{2}y^{27}z^{9}-1241682363288477018167882x^{2}y^{26}z^{10}-34797565664409454022154624x^{2}y^{25}z^{11}-13748130223601093792816557x^{2}y^{24}z^{12}+173406715741801628043633367x^{2}y^{23}z^{13}-3507455485128737678779731x^{2}y^{22}z^{14}-537357056682578738448757375x^{2}y^{21}z^{15}+395044022601310752416005120x^{2}y^{20}z^{16}+764697099442387344468885036x^{2}y^{19}z^{17}-1300905438507958317122202172x^{2}y^{18}z^{18}+106418580620228311737490648x^{2}y^{17}z^{19}+1342000337574765572805001152x^{2}y^{16}z^{20}-1240251170278558841709393877x^{2}y^{15}z^{21}+20350532074457599858577833x^{2}y^{14}z^{22}+748249574754764120479990833x^{2}y^{13}z^{23}-574682132053156710561194266x^{2}y^{12}z^{24}+98653681826118813987776347x^{2}y^{11}z^{25}+128582282270255789194443579x^{2}y^{10}z^{26}-106852972549389484575665173x^{2}y^{9}z^{27}+33071884334354607688386579x^{2}y^{8}z^{28}+1129341689317225280181936x^{2}y^{7}z^{29}-4936451369905992908303016x^{2}y^{6}z^{30}+1951050334758660906814916x^{2}y^{5}z^{31}-350780628018889461141883x^{2}y^{4}z^{32}+10750207137326020225112x^{2}y^{3}z^{33}+8030473919408894009518x^{2}y^{2}z^{34}-1530391814104987575452x^{2}yz^{35}+94993883270163891169x^{2}z^{36}+xy^{37}+25290359xy^{36}z+804527552681xy^{35}z^{2}+1478462323214505xy^{34}z^{3}+533732247881155917xy^{33}z^{4}+60967722961042507143xy^{32}z^{5}+2844076246975452454123xy^{31}z^{6}+62372279613384046211989xy^{30}z^{7}+677601732039951545405191xy^{29}z^{8}+3390630462226673538336467xy^{28}z^{9}+3688593603258633646487312xy^{27}z^{10}-25883397518888242074906283xy^{26}z^{11}-51000612440885740565284880xy^{25}z^{12}+135065918487101712918560862xy^{24}z^{13}+196259546219544306098930519xy^{23}z^{14}-572317540060793252520324338xy^{22}z^{15}-158772086417337957730906452xy^{21}z^{16}+1424278091667375244304057216xy^{20}z^{17}-927776144564427424748644316xy^{19}z^{18}-1331456737074811706332011348xy^{18}z^{19}+2340482182340781792960399044xy^{17}z^{20}-694401241159052394284256152xy^{16}z^{21}-1389851080181250474923262309xy^{15}z^{22}+1638381375631027615175759954xy^{14}z^{23}-491337994146873748341294245xy^{13}z^{24}-414660950426615763624677571xy^{12}z^{25}+487268831755365636570738419xy^{11}z^{26}-190276956259181608307297685xy^{10}z^{27}-9444118166166554716058885xy^{9}z^{28}+45052039038117932495337081xy^{8}z^{29}-21811495055474771209344380xy^{7}z^{30}+4457602591342897279731585xy^{6}z^{31}+241289184986989570740329xy^{5}z^{32}-394735419737706163471515xy^{4}z^{33}+107222451961860252501128xy^{3}z^{34}-14024980868571990893669xy^{2}z^{35}+759156710831978315336xyz^{36}+2544071046558549742xz^{37}+210334y^{37}z+38150429206y^{36}z^{2}+141750921025026y^{35}z^{3}+78979217281644250y^{34}z^{4}+12259068504846596420y^{33}z^{5}+723631642168291296478y^{32}z^{6}+19336625634841574167028y^{31}z^{7}+253667936744741392146306y^{30}z^{8}+1597759867295759522907002y^{29}z^{9}+3447523618891092541001901y^{28}z^{10}-7923179612587466930484452y^{27}z^{11}-34926599936627736571966893y^{26}z^{12}+31786401667957924430982149y^{25}z^{13}+150649487697755414035736305y^{24}z^{14}-173873481023644506116412726y^{23}z^{15}-307273493035158838791034723y^{22}z^{16}+612698854499201235898150129y^{21}z^{17}+52580570383808893866436632y^{20}z^{18}-935699028179766458624861212y^{19}z^{19}+734820107288173369625188852y^{18}z^{20}+291952980655682762878698540y^{17}z^{21}-827140305782794733064297487y^{16}z^{22}+463037165235519474125182498y^{15}z^{23}+95031090105605804402542425y^{14}z^{24}-271325654564702604024464605y^{13}z^{25}+148151563805341662884084734y^{12}z^{26}-14451992438213340406432338y^{11}z^{27}-25713826760911675573963346y^{10}z^{28}+16163254059173078574808560y^{9}z^{29}-4098053064187243602403089y^{8}z^{30}+37656440803421955735490y^{7}z^{31}+295524119992593914322191y^{6}z^{32}-92618767827602963123137y^{5}z^{33}+13168286203423311364551y^{4}z^{34}-761700781878537061124y^{3}z^{35}-2544071046558548999y^{2}z^{36}-yz^{37}}{x^{2}y^{36}-741x^{2}y^{35}z+201967x^{2}y^{34}z^{2}-15133910x^{2}y^{33}z^{3}+191945288x^{2}y^{32}z^{4}+11622940606x^{2}y^{31}z^{5}-374202766390x^{2}y^{30}z^{6}+4531492382218x^{2}y^{29}z^{7}-32969130542949x^{2}y^{28}z^{8}+183845695738947x^{2}y^{27}z^{9}-888919066845531x^{2}y^{26}z^{10}+4278700940452537x^{2}y^{25}z^{11}-21589171156127409x^{2}y^{24}z^{12}+108714481348648913x^{2}y^{23}z^{13}-498550048480467714x^{2}y^{22}z^{14}+1958237867362816193x^{2}y^{21}z^{15}-6341665120877310689x^{2}y^{20}z^{16}+16394789690015820365x^{2}y^{19}z^{17}-32210060332790610278x^{2}y^{18}z^{18}+42583009589794828689x^{2}y^{17}z^{19}-18761757184174431857x^{2}y^{16}z^{20}-69996509060409499305x^{2}y^{15}z^{21}+213388488763989550949x^{2}y^{14}z^{22}-333566046067417589340x^{2}y^{13}z^{23}+334147967171686546361x^{2}y^{12}z^{24}-196667509078915547411x^{2}y^{11}z^{25}+10150130176856186600x^{2}y^{10}z^{26}+112279472847899874063x^{2}y^{9}z^{27}-131237598211670000721x^{2}y^{8}z^{28}+87549131988525406737x^{2}y^{7}z^{29}-38494237731933590050x^{2}y^{6}z^{30}+10561307220458984094x^{2}y^{5}z^{31}-1058689117431640717x^{2}y^{4}z^{32}-466480255126953123x^{2}y^{3}z^{33}+240516662597656251x^{2}y^{2}z^{34}-48446655273437500x^{2}yz^{35}+3814697265625000x^{2}z^{36}+76xy^{36}z-51xy^{35}z^{2}-2856662xy^{34}z^{3}+243382574xy^{33}z^{4}-6049536472xy^{32}z^{5}+20564191234xy^{31}z^{6}+1069309930138xy^{30}z^{7}-18401740303552xy^{29}z^{8}+173113085329218xy^{28}z^{9}-1218205601257295xy^{27}z^{10}+7038400119587981xy^{26}z^{11}-34874730407052263xy^{25}z^{12}+149173036143717080xy^{24}z^{13}-543823809793772243xy^{23}z^{14}+1633535283275040807xy^{22}z^{15}-3775707695989651181xy^{21}z^{16}+5499008389483322001xy^{20}z^{17}+952716149503029491xy^{19}z^{18}-33917039205326815269xy^{18}z^{19}+114580940397552778913xy^{17}z^{20}-235474876665080525162xy^{16}z^{21}+326112660340907082539xy^{15}z^{22}-277670621208893717024xy^{14}z^{23}+48083403018557224362xy^{13}z^{24}+255856052461543108906xy^{12}z^{25}-447006896680293544107xy^{11}z^{26}+426882591037744081295xy^{10}z^{27}-261404798660888652439xy^{9}z^{28}+89519549963378909579xy^{8}z^{29}+5447274322509765255xy^{7}z^{30}-27958429656982421964xy^{6}z^{31}+19066281585693359378xy^{5}z^{32}-7804225921630859374xy^{4}z^{33}+2170429229736328125xy^{3}z^{34}-419044494628906250xy^{2}z^{35}+54550170898437500xyz^{36}-3814697265625000xz^{37}+18y^{37}z-6408y^{36}z^{2}+420869y^{35}z^{3}+22149803y^{34}z^{4}-2290648396y^{33}z^{5}+53730899718y^{32}z^{6}-404847251586y^{31}z^{7}-659568711122y^{30}z^{8}+33319023934164y^{29}z^{9}-339383143196258y^{28}z^{10}+2276621834953389y^{27}z^{11}-11747074715465588y^{26}z^{12}+48454865554660254y^{25}z^{13}-157710664156505155y^{24}z^{14}+371234065904552220y^{23}z^{15}-411806965377471598y^{22}z^{16}-1246948731107691689y^{21}z^{17}+8893039498602776132y^{20}z^{18}-29414995996413107576y^{19}z^{19}+65619315835622591025y^{18}z^{20}-102669024467769017474y^{17}z^{21}+103490061452257586004y^{16}z^{22}-35259229921067903641y^{15}z^{23}-85916904292172239233y^{14}z^{24}+189430841056235096348y^{13}z^{25}-209689330706937005681y^{12}z^{26}+147591023386240007042y^{11}z^{27}-60830578253172590840y^{10}z^{28}+2564482360839405579y^{9}z^{29}+16107705169677639332y^{8}z^{30}-12909356384277331334y^{7}z^{31}+5843223571777347732y^{6}z^{32}-1761302947998047064y^{5}z^{33}+364494323730468656y^{4}z^{34}-50735473632812499y^{3}z^{35}+3814697265625001y^{2}z^{36}}$

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
3.8.0-3.a.1.1 $3$ $16$ $16$ $0$ $0$ full Jacobian
7.16.0-7.a.1.1 $7$ $8$ $8$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
21.64.1-21.a.1.3 $21$ $2$ $2$ $1$ $0$ dimension zero
21.64.1-21.a.1.7 $21$ $2$ $2$ $1$ $0$ 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
21.384.5-21.a.4.3 $21$ $3$ $3$ $5$ $0$ $1^{2}\cdot2$
21.384.5-21.b.1.2 $21$ $3$ $3$ $5$ $0$ $2^{2}$
21.384.5-21.c.4.3 $21$ $3$ $3$ $5$ $0$ $2^{2}$
21.384.9-21.a.2.3 $21$ $3$ $3$ $9$ $0$ $1^{2}\cdot2\cdot4$
21.896.21-21.a.2.3 $21$ $7$ $7$ $21$ $2$ $1^{6}\cdot2^{3}\cdot8$
42.256.5-42.a.3.4 $42$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
42.256.5-42.b.1.4 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.256.5-42.c.3.4 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.256.5-42.d.1.4 $42$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
42.384.9-42.c.1.8 $42$ $3$ $3$ $9$ $0$ $1^{4}\cdot4$
63.384.5-63.a.1.1 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.b.1.1 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.c.1.3 $63$ $3$ $3$ $5$ $2$ $2^{2}$
63.384.5-63.d.1.1 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.e.1.3 $63$ $3$ $3$ $5$ $4$ $2^{2}$
63.384.5-63.f.1.1 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.9-63.a.1.1 $63$ $3$ $3$ $9$ $0$ $1^{2}\cdot2\cdot4$
63.384.9-63.b.2.3 $63$ $3$ $3$ $9$ $2$ $1^{4}\cdot4$
63.384.13-63.a.2.2 $63$ $3$ $3$ $13$ $0$ $1^{2}\cdot2^{3}\cdot4$
84.256.5-84.c.1.12 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.h.1.8 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.k.1.8 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.n.1.8 $84$ $2$ $2$ $5$ $?$ not computed
84.512.17-84.l.1.15 $84$ $4$ $4$ $17$ $?$ not computed
168.256.5-168.f.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.l.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.t.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.z.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bf.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bl.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.br.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bx.1.15 $168$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.e.4.1 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.f.2.7 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.k.4.1 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.l.1.8 $210$ $2$ $2$ $5$ $?$ not computed
273.384.5-273.j.4.6 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.k.1.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.l.4.6 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.m.3.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.n.1.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.o.4.6 $273$ $3$ $3$ $5$ $?$ not computed