Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $12^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ 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: | 12Q1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.48.1.36 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&10\\2&9\end{bmatrix}$, $\begin{bmatrix}3&10\\11&9\end{bmatrix}$, $\begin{bmatrix}10&9\\9&2\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $D_6:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $6$ |
| Cyclic 12-torsion field degree: | $24$ |
| Full 12-torsion field degree: | $96$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 4x - 4 $ |
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 | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(-2:0:1)$, $(0:1:0)$ | ||
| 6534.g1 | no | $\tfrac{3375}{64}$ | $= 2^{-6} \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(-1:0:1)$, $(2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6}\cdot\frac{96x^{2}y^{22}-11864x^{2}y^{21}z+600097x^{2}y^{20}z^{2}-17753016x^{2}y^{19}z^{3}+373042772x^{2}y^{18}z^{4}-5842363464x^{2}y^{17}z^{5}+72252778086x^{2}y^{16}z^{6}-753120031168x^{2}y^{15}z^{7}+6186782246944x^{2}y^{14}z^{8}-43537077042144x^{2}y^{13}z^{9}+302285717682773x^{2}y^{12}z^{10}-1274207583356296x^{2}y^{11}z^{11}+5967406517591100x^{2}y^{10}z^{12}-39555755523103640x^{2}y^{9}z^{13}-46978373194292178x^{2}y^{8}z^{14}-922786560813112416x^{2}y^{7}z^{15}-2365802847493058224x^{2}y^{6}z^{16}-9563160812727561464x^{2}y^{5}z^{17}-20451275821111204533x^{2}y^{4}z^{18}-38193055492294844552x^{2}y^{3}z^{19}-54432436498477380116x^{2}y^{2}z^{20}-39031121279833499148x^{2}yz^{21}-10271347698613171691x^{2}z^{22}-12xy^{23}+2826xy^{22}z-181408xy^{21}z^{2}+6517746xy^{20}z^{3}-153936852xy^{19}z^{4}+2730973870xy^{18}z^{5}-37684683600xy^{17}z^{6}+413993388048xy^{16}z^{7}-3956863597808xy^{15}z^{8}+30562025874696xy^{14}z^{9}-189990616849920xy^{13}z^{10}+1292372131856734xy^{12}z^{11}-5469966268851348xy^{11}z^{12}+18427172660772234xy^{10}z^{13}-168615952313331568xy^{9}z^{14}-273680741737013208xy^{8}z^{15}-3424061688592629180xy^{7}z^{16}-8584729593948483422xy^{6}z^{17}-31714125436859735904xy^{5}z^{18}-65848254932768933826xy^{4}z^{19}-117831759723517566460xy^{3}z^{20}-164153255466083872263xy^{2}z^{21}-117093364018396004352xyz^{22}-30814043166352605184xz^{23}+y^{24}-556y^{23}z+48358y^{22}z^{2}-1975636y^{21}z^{3}+52410297y^{20}z^{4}-982400556y^{19}z^{5}+14287861942y^{18}z^{6}-168257772752y^{17}z^{7}+1563209272252y^{16}z^{8}-12851362130224y^{15}z^{9}+91902491542872y^{14}z^{10}-450081170817052y^{13}z^{11}+2758402865651245y^{12}z^{12}-13218532929487740y^{11}z^{13}+7999230153995922y^{10}z^{14}-372368456070513024y^{9}z^{15}-863715906047978123y^{8}z^{16}-5362962149195085884y^{7}z^{17}-12539279516029882946y^{6}z^{18}-34520584507891600748y^{5}z^{19}-63446019273029494973y^{4}z^{20}-92649077890529206244y^{3}z^{21}-113144601690664421801y^{2}z^{22}-78062242711299581904yz^{23}-20542695469533046868z^{24}}{z^{4}(26x^{2}y^{18}+41379x^{2}y^{16}z^{2}+12510664x^{2}y^{14}z^{4}+1426581124x^{2}y^{12}z^{6}+77970031704x^{2}y^{10}z^{8}+2279759363325x^{2}y^{8}z^{10}+37319608484820x^{2}y^{6}z^{12}+340352871222240x^{2}y^{4}z^{14}+1606218984731754x^{2}y^{2}z^{16}+3043362286160901x^{2}z^{18}+421xy^{18}z+351888xy^{16}z^{3}+78208539xy^{14}z^{5}+7303328012xy^{12}z^{7}+345089520612xy^{10}z^{9}+9015643835280xy^{8}z^{11}+134866721104119xy^{6}z^{13}+1142581761068148xy^{4}z^{15}+5072270477289129xy^{2}z^{17}+9130086859014144xz^{19}+y^{20}+4651y^{18}z^{2}+2177656y^{16}z^{4}+332729551y^{14}z^{6}+23011550716y^{12}z^{8}+833237210988y^{10}z^{10}+16945709793750y^{8}z^{12}+198278255240487y^{6}z^{14}+1309455938336553y^{4}z^{16}+4480505587058607y^{2}z^{18}+6086724573384684z^{20})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.12.0.p.1 | $12$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 12.24.1.o.1 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.96.5.b.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.96.5.c.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.96.5.e.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.96.5.g.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.bg.1 | $12$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.96.5.dp.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.96.5.ds.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.fe.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.fk.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.192.9.ry.1 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
| 36.144.5.q.1 | $36$ | $3$ | $3$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 36.144.7.p.1 | $36$ | $3$ | $3$ | $7$ | $3$ | $1^{4}\cdot2$ |
| 36.144.7.r.1 | $36$ | $3$ | $3$ | $7$ | $5$ | $1^{4}\cdot2$ |
| 60.96.5.co.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.96.5.cp.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.96.5.cs.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.96.5.ct.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.240.17.ys.1 | $60$ | $5$ | $5$ | $17$ | $8$ | $1^{16}$ |
| 60.288.17.nk.1 | $60$ | $6$ | $6$ | $17$ | $2$ | $1^{16}$ |
| 60.480.33.yp.1 | $60$ | $10$ | $10$ | $33$ | $15$ | $1^{32}$ |
| 84.96.5.co.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.96.5.cp.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.96.5.cs.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.96.5.ct.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.5.bfq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.5.bft.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.5.bgc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.5.bgf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.96.5.r.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.96.5.s.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.96.5.v.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.96.5.w.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.96.5.q.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.96.5.r.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.96.5.u.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.96.5.v.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.96.5.beo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.96.5.ber.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.96.5.bfa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.96.5.bfd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.96.5.q.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.96.5.r.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.96.5.u.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.96.5.v.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.96.5.q.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.96.5.r.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.96.5.u.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.96.5.v.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.96.5.yq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.96.5.yt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.96.5.zc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.96.5.zf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.96.5.q.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.96.5.r.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.96.5.u.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.96.5.v.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.5.za.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.5.zd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.5.zm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.5.zp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |