Invariants
| Level: | $21$ | $\SL_2$-level: | $21$ | Newform level: | $21$ | ||
| Index: | $64$ | $\PSL_2$-index: | $32$ | ||||
| Genus: | $1 = 1 + \frac{ 32 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot3\cdot7\cdot21$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 21B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 21.64.1.8 |
Level structure
| $\GL_2(\Z/21\Z)$-generators: | $\begin{bmatrix}2&14\\0&10\end{bmatrix}$, $\begin{bmatrix}5&15\\0&19\end{bmatrix}$, $\begin{bmatrix}17&15\\0&2\end{bmatrix}$ |
| $\GL_2(\Z/21\Z)$-subgroup: | $C_{42}:C_6^2$ |
| Contains $-I$: | no $\quad$ (see 21.32.1.a.1 for the level structure with $-I$) |
| Cyclic 21-isogeny field degree: | $1$ |
| Cyclic 21-torsion field degree: | $12$ |
| Full 21-torsion field degree: | $1512$ |
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} - 4x - 1 $ |
Rational points
This modular curve has rational points, including 4 rational_cusps and 4 known non-cuspidal non-CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(-1:-1:1)$, $(-2:1:1)$, $(0:1:0)$, $(5:8:1)$ | ||
| 162.b4 | no | $\tfrac{3375}{2}$ | $= 2^{-1} \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(2:-1:1)$ |
| 162.b3 | no | $\tfrac{-140625}{8}$ | $= -1 \cdot 2^{-3} \cdot 3^{2} \cdot 5^{6}$ | $11.854$ | $(-1:2:1)$ |
| 162.b2 | no | $\tfrac{-1159088625}{2097152}$ | $= -1 \cdot 2^{-21} \cdot 3^{2} \cdot 5^{3} \cdot 101^{3}$ | $20.871$ | $(5:-13:1)$ |
| 162.b1 | no | $\tfrac{-189613868625}{128}$ | $= -1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$ | $25.968$ | $(-1/4:1/8:1)$ |
Maps to other modular curves
$j$-invariant map of degree 32 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(y-2z)^{3}(23x^{2}y^{14}+377x^{2}y^{13}z+4095x^{2}y^{12}z^{2}+245068x^{2}y^{11}z^{3}+2435478x^{2}y^{10}z^{4}+29457995x^{2}y^{9}z^{5}+746040474x^{2}y^{8}z^{6}+6047469528x^{2}y^{7}z^{7}+53287573992x^{2}y^{6}z^{8}+717132085280x^{2}y^{5}z^{9}+4961101986080x^{2}y^{4}z^{10}+18103312690368x^{2}y^{3}z^{11}+39619021970944x^{2}y^{2}z^{12}+51911470190592x^{2}yz^{13}+33972342259712x^{2}z^{14}+6xy^{15}+204xy^{14}z+1280xy^{13}z^{2}+87098xy^{12}z^{3}+1577041xy^{11}z^{4}+11660113xy^{10}z^{5}+328331220xy^{9}z^{6}+3991172196xy^{8}z^{7}+28675924416xy^{7}z^{8}+377920774704xy^{6}z^{9}+3489741498816xy^{5}z^{10}+17183031111680xy^{4}z^{11}+51491907143168xy^{3}z^{12}+98709014081536xy^{2}z^{13}+117416508030976xyz^{14}+71848372469760xz^{15}+y^{16}+66y^{15}z+384y^{14}z^{2}+18499y^{13}z^{3}+562283y^{12}z^{4}+3281507y^{11}z^{5}+86201533y^{10}z^{6}+1428256201y^{9}z^{7}+8736970218y^{8}z^{8}+112307562952y^{7}z^{9}+1234256972808y^{6}z^{10}+6151243031328y^{5}z^{11}+16793531623264y^{4}z^{12}+28881347449664y^{3}z^{13}+31632474120704y^{2}z^{14}+22842692497408yz^{15}+16062921539584z^{16})}{z^{2}(8x^{2}y^{15}+364x^{2}y^{14}z+974x^{2}y^{13}z^{2}-44891x^{2}y^{12}z^{3}-365134x^{2}y^{11}z^{4}+535048x^{2}y^{10}z^{5}+19925700x^{2}y^{9}z^{6}+103175658x^{2}y^{8}z^{7}+71666946x^{2}y^{7}z^{8}-2146829497x^{2}y^{6}z^{9}-13917777272x^{2}y^{5}z^{10}-54067645471x^{2}y^{4}z^{11}-131663720534x^{2}y^{3}z^{12}-248391467944x^{2}y^{2}z^{13}-275148358478x^{2}yz^{14}-241660589493x^{2}z^{15}+xy^{16}+148xy^{15}z+1556xy^{14}z^{2}-14273xy^{13}z^{3}-254871xy^{12}z^{4}-655706xy^{11}z^{5}+9157729xy^{10}z^{6}+84905626xy^{9}z^{7}+238664745xy^{8}z^{8}-687953450xy^{7}z^{9}-10066438046xy^{6}z^{10}-48914984194xy^{5}z^{11}-161327953799xy^{4}z^{12}-355811912190xy^{3}z^{13}-610320252041xy^{2}z^{14}-636177787013xyz^{15}-511089803198xz^{16}+36y^{16}z+672y^{15}z^{2}-3053y^{14}z^{3}-94747y^{13}z^{4}-324723y^{12}z^{5}+3056786y^{11}z^{6}+31672288y^{10}z^{7}+94932910y^{9}z^{8}-284015448y^{8}z^{9}-3493289259y^{7}z^{10}-17645093604y^{6}z^{11}-50977456313y^{5}z^{12}-117062543890y^{4}z^{13}-165359854062y^{3}z^{14}-213998780587y^{2}z^{15}-117967467713yz^{16}-114265321181z^{17})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.16.0-7.a.1.1 | $21$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.128.1-21.a.1.3 | $21$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 21.128.1-21.a.2.4 | $21$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 21.128.1-21.a.3.1 | $21$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 21.128.1-21.a.4.2 | $21$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 21.192.3-21.a.1.5 | $21$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 21.192.3-21.a.2.5 | $21$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 21.192.3-21.b.1.2 | $21$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 21.192.5-21.a.1.5 | $21$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 21.448.11-21.a.1.3 | $21$ | $7$ | $7$ | $11$ | $2$ | $1^{6}\cdot2^{2}$ |
| 42.128.3-42.a.1.6 | $42$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 42.128.3-42.b.1.4 | $42$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 42.128.3-42.c.1.6 | $42$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 42.128.3-42.d.1.3 | $42$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 42.128.3-42.e.1.4 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.128.3-42.e.2.3 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.128.3-42.e.3.4 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.128.3-42.e.4.2 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.192.5-42.a.1.8 | $42$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 63.192.3-63.a.1.2 | $63$ | $3$ | $3$ | $3$ | $2$ | $2$ |
| 63.192.3-63.a.2.1 | $63$ | $3$ | $3$ | $3$ | $2$ | $2$ |
| 63.192.3-63.b.1.8 | $63$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 63.192.3-63.b.2.5 | $63$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 63.192.3-63.c.1.6 | $63$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 63.192.3-63.c.2.6 | $63$ | $3$ | $3$ | $3$ | $0$ | $2$ |
| 63.192.5-63.a.1.7 | $63$ | $3$ | $3$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 63.192.5-63.b.1.6 | $63$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 63.192.7-63.a.1.6 | $63$ | $3$ | $3$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 84.128.1-84.c.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.128.1-84.c.2.7 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.128.1-84.c.3.5 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.128.1-84.c.4.11 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.128.3-84.a.1.13 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.b.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.e.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.f.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.g.1.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.g.2.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.g.3.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.128.3-84.g.4.6 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.256.9-84.a.1.16 | $84$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 105.128.1-105.a.1.6 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.128.1-105.a.2.6 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.128.1-105.a.3.4 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.128.1-105.a.4.4 | $105$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 105.320.11-105.a.1.2 | $105$ | $5$ | $5$ | $11$ | $?$ | not computed |
| 105.384.13-105.a.1.9 | $105$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 147.448.11-147.a.1.3 | $147$ | $7$ | $7$ | $11$ | $?$ | not computed |
| 168.128.1-168.c.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.c.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.c.3.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.c.4.23 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.d.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.d.2.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.d.3.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.1-168.d.4.27 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.128.3-168.a.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.b.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.c.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.d.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.g.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.h.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.i.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.j.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.k.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.k.2.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.k.3.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.k.4.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.l.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.l.2.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.l.3.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.128.3-168.l.4.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.a.1.8 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.b.1.4 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.c.1.8 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.d.1.3 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.e.1.8 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.e.2.6 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.e.3.6 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.128.3-210.e.4.5 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 231.128.1-231.a.1.8 | $231$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 231.128.1-231.a.2.8 | $231$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 231.128.1-231.a.3.4 | $231$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 231.128.1-231.a.4.4 | $231$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 273.128.1-273.a.1.2 | $273$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 273.128.1-273.a.2.1 | $273$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 273.128.1-273.a.3.5 | $273$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 273.128.1-273.a.4.1 | $273$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 273.192.3-273.a.1.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 273.192.3-273.a.2.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 273.192.3-273.b.1.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 273.192.3-273.b.2.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 273.192.3-273.c.1.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 273.192.3-273.c.2.3 | $273$ | $3$ | $3$ | $3$ | $?$ | not computed |