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Properties

Label 21.128.1-21.a.2.4
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: 21.128.1.14

Level structure

$\GL_2(\Z/21\Z)$-generators: $\begin{bmatrix}10&3\\0&5\end{bmatrix}$, $\begin{bmatrix}11&8\\0&10\end{bmatrix}$
$\GL_2(\Z/21\Z)$-subgroup: $C_{21}:C_6^2$
Contains $-I$: no $\quad$ (see 21.64.1.a.2 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$$(1:1:1)$, $(1:-2:1)$
162.b4 no$\tfrac{3375}{2}$ $= 2^{-1} \cdot 3^{3} \cdot 5^{3}$$8.124$$(0:0:1)$, $(0:1:0)$

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 3^3\,\frac{(y^{2}+yz+z^{2})^{3}(21930x^{2}y^{30}-1976691x^{2}y^{29}z+59389632x^{2}y^{28}z^{2}-802146813x^{2}y^{27}z^{3}+1730956871x^{2}y^{26}z^{4}+58337023202x^{2}y^{25}z^{5}-309569755963x^{2}y^{24}z^{6}-1264491374900x^{2}y^{23}z^{7}+5885307275746x^{2}y^{22}z^{8}+19041889869403x^{2}y^{21}z^{9}-9853523962734x^{2}y^{20}z^{10}-48301649335071x^{2}y^{19}z^{11}+20166022980198x^{2}y^{18}z^{12}+61421833260155x^{2}y^{17}z^{13}-38479054005100x^{2}y^{16}z^{14}-40137621604600x^{2}y^{15}z^{15}+40754576459788x^{2}y^{14}z^{16}+8076914737063x^{2}y^{13}z^{17}-22095813350384x^{2}y^{12}z^{18}+4721835306675x^{2}y^{11}z^{19}+5408069207226x^{2}y^{10}z^{20}-3009517691493x^{2}y^{9}z^{21}-238590175504x^{2}y^{8}z^{22}+582300085142x^{2}y^{7}z^{23}-116122899340x^{2}y^{6}z^{24}-36353309618x^{2}y^{5}z^{25}+16020735046x^{2}y^{4}z^{26}-424276268x^{2}y^{3}z^{27}-590812575x^{2}y^{2}z^{28}+42955221x^{2}yz^{29}+3765213x^{2}z^{30}+4225xy^{31}-533326xy^{30}z+18559305xy^{29}z^{2}-133345655xy^{28}z^{3}-2726601096xy^{27}z^{4}+41567043953xy^{26}z^{5}-44423450280xy^{25}z^{6}-1412648692990xy^{24}z^{7}+1902917881342xy^{23}z^{8}+20374890538974xy^{22}z^{9}+7961890720761xy^{21}z^{10}-58355085081729xy^{20}z^{11}-15748395972135xy^{19}z^{12}+94933883208220xy^{18}z^{13}-6624572095726xy^{17}z^{14}-90975648530988xy^{16}z^{15}+38533273115660xy^{15}z^{16}+44730339493180xy^{14}z^{17}-37894626995613xy^{13}z^{18}-5418555913443xy^{12}z^{19}+15984842641329xy^{11}z^{20}-4012773016608xy^{10}z^{21}-2643194176220xy^{9}z^{22}+1651486796834xy^{8}z^{23}-36904479579xy^{7}z^{24}-206213744224xy^{6}z^{25}+51146798386xy^{5}z^{26}+6021178340xy^{4}z^{27}-3750006629xy^{3}z^{28}+224127153xy^{2}z^{29}+67504514xyz^{30}-2670734xz^{31}+250y^{32}-13812y^{31}z+368180y^{30}z^{2}+41277676y^{29}z^{3}-1580987644y^{28}z^{4}+13071774314y^{27}z^{5}+32580446556y^{26}z^{6}-613170004916y^{25}z^{7}-247418710378y^{24}z^{8}+9318995120028y^{23}z^{9}+12906201079581y^{22}z^{10}-21326852332155y^{21}z^{11}-28443104041788y^{20}z^{12}+34614513329222y^{19}z^{13}+26023479489946y^{18}z^{14}-39044339366721y^{17}z^{15}-7114973838560y^{16}z^{16}+26017475866208y^{15}z^{17}-5373746831169y^{14}z^{18}-8794422296511y^{13}z^{19}+4694758015944y^{12}z^{20}+920326648182y^{11}z^{21}-1323157827766y^{10}z^{22}+194986235641y^{9}z^{23}+139396981566y^{8}z^{24}-52181751896y^{7}z^{25}-2067857911y^{6}z^{26}+3410137259y^{5}z^{27}-292935504y^{4}z^{28}-64827872y^{3}z^{29}+2682404y^{2}z^{30}+225yz^{31}-250z^{32})}{(y-z)^{7}(681x^{2}y^{29}+60618x^{2}y^{28}z+336798x^{2}y^{27}z^{2}-13220823x^{2}y^{26}z^{3}-78967504x^{2}y^{25}z^{4}+751585786x^{2}y^{24}z^{5}+6039361776x^{2}y^{23}z^{6}-3126811610x^{2}y^{22}z^{7}-141690534028x^{2}y^{21}z^{8}-512703422910x^{2}y^{20}z^{9}-740354659338x^{2}y^{19}z^{10}-187145537691x^{2}y^{18}z^{11}+656318583438x^{2}y^{17}z^{12}+516613730192x^{2}y^{16}z^{13}-262030625219x^{2}y^{15}z^{14}-337712953425x^{2}y^{14}z^{15}+75180818440x^{2}y^{13}z^{16}+124581931586x^{2}y^{12}z^{17}-22018009635x^{2}y^{11}z^{18}-28441614408x^{2}y^{10}z^{19}+5924690424x^{2}y^{9}z^{20}+3800667588x^{2}y^{8}z^{21}-1026489067x^{2}y^{7}z^{22}-249799046x^{2}y^{6}z^{23}+90847281x^{2}y^{5}z^{24}+4551130x^{2}y^{4}z^{25}-3259324x^{2}y^{3}z^{26}+100911x^{2}y^{2}z^{27}+28932x^{2}yz^{28}-996x^{2}z^{29}+76xy^{30}+19218xy^{29}z+365613xy^{28}z^{2}-4148669xy^{27}z^{3}-69886712xy^{26}z^{4}+175890834xy^{25}z^{5}+4401907020xy^{24}z^{6}+7911133226xy^{23}z^{7}-81261748206xy^{22}z^{8}-443754596118xy^{21}z^{9}-859843511646xy^{20}z^{10}-468805403493xy^{19}z^{11}+689411410590xy^{18}z^{12}+905252183269xy^{17}z^{13}-186749168967xy^{16}z^{14}-627925171680xy^{15}z^{15}+8902288904xy^{14}z^{16}+266213349360xy^{13}z^{17}-2089579884xy^{12}z^{18}-74688783501xy^{11}z^{19}+5437738602xy^{10}z^{20}+13227618456xy^{9}z^{21}-2079694112xy^{8}z^{22}-1302188715xy^{7}z^{23}+316459458xy^{6}z^{24}+54490478xy^{5}z^{25}-19775961xy^{4}z^{26}-185557xy^{3}z^{27}+395574xy^{2}z^{28}-15900xyz^{29}-1000xz^{30}+4y^{31}+3850y^{30}z+142212y^{29}z^{2}-548195y^{28}z^{3}-27931638y^{27}z^{4}-20819197y^{26}z^{5}+1659560714y^{25}z^{6}+6196910274y^{24}z^{7}-23196078724y^{23}z^{8}-205190608882y^{22}z^{9}-539942759392y^{21}z^{10}-584117691430y^{20}z^{11}+3209364263y^{19}z^{12}+530604834991y^{18}z^{13}+249206241755y^{17}z^{14}-224997545437y^{16}z^{15}-155814843954y^{15}z^{16}+67806364085y^{14}z^{17}+49405777673y^{13}z^{18}-16676421928y^{12}z^{19}-8971348744y^{11}z^{20}+3080314370y^{10}z^{21}+861159475y^{9}z^{22}-349561072y^{8}z^{23}-31969918y^{7}z^{24}+19423614y^{6}z^{25}-219514y^{5}z^{26}-377051y^{4}z^{27}+17212y^{3}z^{28}+1109y^{2}z^{29}+28yz^{30}+4z^{31})}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
21.64.1-21.a.1.2 $21$ $2$ $2$ $1$ $0$ dimension zero
21.64.1-21.a.1.8 $21$ $2$ $2$ $1$ $0$ 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.2.4 $21$ $3$ $3$ $5$ $0$ $1^{2}\cdot2$
21.384.5-21.b.4.1 $21$ $3$ $3$ $5$ $0$ $2^{2}$
21.384.5-21.c.3.2 $21$ $3$ $3$ $5$ $0$ $2^{2}$
21.384.9-21.a.1.4 $21$ $3$ $3$ $9$ $0$ $1^{2}\cdot2\cdot4$
21.896.21-21.a.2.1 $21$ $7$ $7$ $21$ $2$ $1^{6}\cdot2^{3}\cdot8$
42.256.5-42.a.4.1 $42$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
42.256.5-42.b.3.1 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.256.5-42.c.4.1 $42$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
42.256.5-42.d.2.3 $42$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
42.384.9-42.c.3.5 $42$ $3$ $3$ $9$ $0$ $1^{4}\cdot4$
63.384.5-63.a.3.4 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.b.3.4 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.c.2.2 $63$ $3$ $3$ $5$ $2$ $2^{2}$
63.384.5-63.d.3.4 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.5-63.e.2.2 $63$ $3$ $3$ $5$ $4$ $2^{2}$
63.384.5-63.f.2.4 $63$ $3$ $3$ $5$ $0$ $2^{2}$
63.384.9-63.a.3.4 $63$ $3$ $3$ $9$ $0$ $1^{2}\cdot2\cdot4$
63.384.9-63.b.3.4 $63$ $3$ $3$ $9$ $2$ $1^{4}\cdot4$
63.384.13-63.a.1.3 $63$ $3$ $3$ $13$ $0$ $1^{2}\cdot2^{3}\cdot4$
84.256.5-84.c.3.9 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.h.3.5 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.k.3.5 $84$ $2$ $2$ $5$ $?$ not computed
84.256.5-84.n.3.5 $84$ $2$ $2$ $5$ $?$ not computed
84.512.17-84.l.3.10 $84$ $4$ $4$ $17$ $?$ not computed
168.256.5-168.f.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.l.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.t.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.z.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bf.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bl.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.br.3.10 $168$ $2$ $2$ $5$ $?$ not computed
168.256.5-168.bx.3.10 $168$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.e.3.6 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.f.1.5 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.k.3.6 $210$ $2$ $2$ $5$ $?$ not computed
210.256.5-210.l.2.5 $210$ $2$ $2$ $5$ $?$ not computed
273.384.5-273.j.1.4 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.k.3.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.l.1.6 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.m.2.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.n.3.7 $273$ $3$ $3$ $5$ $?$ not computed
273.384.5-273.o.1.4 $273$ $3$ $3$ $5$ $?$ not computed