Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (none of which are rational) | Cusp widths | $12^{2}$ | Cusp orbits | $2$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 12G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.1.21 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}2&7\\7&10\end{bmatrix}$, $\begin{bmatrix}8&9\\3&8\end{bmatrix}$, $\begin{bmatrix}9&11\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_{24}:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $24$ |
| Cyclic 12-torsion field degree: | $96$ |
| Full 12-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 219x + 1190 $ |
Rational points
This modular curve has rational points, including 1 rational CM point and 1 known non-cuspidal non-CM point, but no rational cusps. 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 | |
|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(7:0:1)$, $(0:1:0)$ |
| 6912.i1 | no | $-21024576$ | $= -1 \cdot 2^{6} \cdot 3^{3} \cdot 23^{3}$ | $16.861$ | $(-17:0:1)$, $(10:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{240x^{2}y^{14}-89424x^{2}y^{13}z-56377458x^{2}y^{12}z^{2}+19842633504x^{2}y^{11}z^{3}-897602274360x^{2}y^{10}z^{4}-305704376768736x^{2}y^{9}z^{5}+41018949423409455x^{2}y^{8}z^{6}-720799730339954400x^{2}y^{7}z^{7}-189140255180282763000x^{2}y^{6}z^{8}+14212919270382916559184x^{2}y^{5}z^{9}-188962395376576243695129x^{2}y^{4}z^{10}-20880843119794954455444432x^{2}y^{3}z^{11}+1156450187744314713861201612x^{2}y^{2}z^{12}-24325205512057119789789734064x^{2}yz^{13}+193666059741542665227516814599x^{2}z^{14}-25140xy^{14}z+8331984xy^{13}z^{2}+1416408120xy^{12}z^{3}-667360495584xy^{11}z^{4}+48173970450897xy^{10}z^{5}+5664595749438384xy^{9}z^{6}-976482233058748878xy^{8}z^{7}+29305426209508689696xy^{7}z^{8}+3148301541429953693712xy^{6}z^{9}-278401403588026051036944xy^{5}z^{10}+5033319315033119292719136xy^{4}z^{11}+317435517252846529769057328xy^{3}z^{12}-19360785683964301711671147057xy^{2}z^{13}+413528494316645588685662785824xyz^{14}-3292323008528021512393142144442xz^{15}-y^{16}+432y^{15}z+1478688y^{14}z^{2}-472493088y^{13}z^{3}-3943721844y^{12}z^{4}+14679325748448y^{11}z^{5}-1549241199147456y^{10}z^{6}-19347777978269040y^{9}z^{7}+12613691353697674092y^{8}z^{8}-693965420963722463472y^{7}z^{9}-8091372687365906916624y^{6}z^{10}+2187585057794914083055968y^{5}z^{11}-74678367708833164640358066y^{4}z^{12}-185337092198883845416236048y^{3}z^{13}+70790021800250715482463540216y^{2}z^{14}-1702764390125720251099942531632yz^{15}+13556624112952150028838328289727z^{16}}{24x^{2}y^{14}-460242x^{2}y^{12}z^{2}+3530500344x^{2}y^{10}z^{4}-14776818628329x^{2}y^{8}z^{6}+36853620447442464x^{2}y^{6}z^{8}-54910635783378022809x^{2}y^{4}z^{10}+45266322610785247776324x^{2}y^{2}z^{12}-15917052660496352994541185x^{2}z^{14}-732xy^{14}z+11164392xy^{12}z^{3}-76681602207xy^{10}z^{5}+298409480432946xy^{8}z^{7}-703535860088838216xy^{6}z^{9}+1000870829186004453384xy^{4}z^{11}-793967698959155639185545xy^{2}z^{13}+270596364648393710922469110xz^{15}-y^{16}+19824y^{14}z^{2}-178267716y^{12}z^{4}+900453002616y^{10}z^{6}-2766458725574364y^{8}z^{8}+5291309733819788160y^{6}z^{10}-6161636348917657750170y^{4}z^{12}+4002485342862296198721168y^{2}z^{14}-1114258380584396445067571721z^{16}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.6.0.d.1 | $12$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 12.12.0.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.48.3.d.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.f.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.l.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.48.3.n.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.do.1 | $12$ | $3$ | $3$ | $3$ | $0$ | $1^{2}$ |
| 24.48.3.k.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.q.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bi.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.3.bo.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.5.iw.1 | $24$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
| 36.72.3.y.1 | $36$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 36.216.15.eb.1 | $36$ | $9$ | $9$ | $15$ | $10$ | $1^{4}\cdot2^{5}$ |
| 60.48.3.bh.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bj.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bl.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.48.3.bn.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.120.9.it.1 | $60$ | $5$ | $5$ | $9$ | $6$ | $1^{8}$ |
| 60.144.9.ll.1 | $60$ | $6$ | $6$ | $9$ | $3$ | $1^{8}$ |
| 60.240.17.yb.1 | $60$ | $10$ | $10$ | $17$ | $12$ | $1^{16}$ |
| 84.48.3.t.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.v.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.x.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.48.3.z.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.15.bt.1 | $84$ | $8$ | $8$ | $15$ | $?$ | not computed |
| 120.48.3.dm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ds.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.dy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.3.ee.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.t.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.v.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.x.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.48.3.z.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.288.23.bv.1 | $132$ | $12$ | $12$ | $23$ | $?$ | not computed |
| 156.48.3.t.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.v.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.x.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.48.3.z.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.cs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.cy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.de.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.48.3.dk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.t.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.v.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.x.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.48.3.z.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.t.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.v.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.x.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.48.3.z.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.cs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.cy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.de.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.3.dk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.t.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.v.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.x.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.48.3.z.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.cs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.cy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.de.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.3.dk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |