Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ 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 $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.96.1.94 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&5\\0&7\end{bmatrix}$, $\begin{bmatrix}5&8\\6&11\end{bmatrix}$, $\begin{bmatrix}7&9\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_6:D_4$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.o.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| Full 12-torsion field degree: | $48$ |
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 1 rational CM point but no rational cusps or other known rational 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 | |
|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(-2:0:1)$, $(-1:0:1)$, $(2:0:1)$, $(0:1:0)$ | |
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 2^{14}\cdot3^3\,\frac{zy^{3}(10x^{2}y^{18}-736x^{2}y^{17}z+22521x^{2}y^{16}z^{2}-358728x^{2}y^{15}z^{3}+2781864x^{2}y^{14}z^{4}-814176x^{2}y^{13}z^{5}-172920424x^{2}y^{12}z^{6}+1417448524x^{2}y^{11}z^{7}-5826164904x^{2}y^{10}z^{8}+33645519104x^{2}y^{9}z^{9}-253889232493x^{2}y^{8}z^{10}+1009427991420x^{2}y^{7}z^{11}-3060016345436x^{2}y^{6}z^{12}+16138912702352x^{2}y^{5}z^{13}-41979023814276x^{2}y^{4}z^{14}+68357139681740x^{2}y^{3}z^{15}-410141445663110x^{2}y^{2}z^{16}-1283918200896377x^{2}z^{18}-25xy^{18}z+1104xy^{17}z^{2}-11256xy^{16}z^{3}-179088xy^{15}z^{4}+4206645xy^{14}z^{5}-4892508xy^{13}z^{6}-558507488xy^{12}z^{7}+6296584752xy^{11}z^{8}-31984123140xy^{10}z^{9}+143865810592xy^{9}z^{10}-940892663100xy^{8}z^{11}+4093361403528xy^{7}z^{12}-11459434632751xy^{6}z^{13}+54116794688700xy^{5}z^{14}-154918400876424xy^{4}z^{15}+205069924433920xy^{3}z^{16}-1337414159674413xy^{2}z^{17}-3851755662082048xz^{19}-y^{20}+92y^{19}z-3751y^{18}z^{2}+90832y^{17}z^{3}-1435992y^{16}z^{4}+14904456y^{15}z^{5}-95890979y^{14}z^{6}+355234060y^{13}z^{7}-1318659676y^{12}z^{8}+13337497332y^{11}z^{9}-93057880540y^{10}z^{10}+344878976160y^{9}z^{11}-1517760503142y^{8}z^{12}+7831841864180y^{7}z^{13}-19504002535899y^{6}z^{14}+60762966629636y^{5}z^{15}-237763963224665y^{4}z^{16}+136712833644752y^{3}z^{17}-1355245848604091y^{2}z^{18}-2567837446836764z^{20})}{12x^{2}y^{22}-2561x^{2}y^{20}z^{2}+182896x^{2}y^{18}z^{4}+387882x^{2}y^{16}z^{6}-1015428912x^{2}y^{14}z^{8}+36210315435x^{2}y^{12}z^{10}+3388653456360x^{2}y^{10}z^{12}-105467029420318x^{2}y^{8}z^{14}-7385426436654324x^{2}y^{6}z^{16}-127749987690690027x^{2}y^{4}z^{18}-893607185213181280x^{2}y^{2}z^{20}-2218611123791166013x^{2}z^{22}-42xy^{22}z+3810xy^{20}z^{3}+92378xy^{18}z^{5}+2302992xy^{16}z^{7}-4535297352xy^{14}z^{9}+289224461982xy^{12}z^{11}+10075944029550xy^{10}z^{13}-642872749839864xy^{8}z^{15}-29949279572534562xy^{6}z^{17}-448729386849568722xy^{4}z^{19}-2865706083783071553xy^{2}z^{21}-6655833303041441792xz^{23}-y^{24}+358y^{22}z^{2}-41669y^{20}z^{4}+3870230y^{18}z^{6}-254058972y^{16}z^{8}-2077810344y^{14}z^{10}+1163457082655y^{12}z^{12}-5106390518758y^{10}z^{14}-2291956414568165y^{8}z^{16}-58442153031889442y^{6}z^{18}-598305851844701719y^{4}z^{20}-2711635915506578239y^{2}z^{22}-4437222179637696268z^{24}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-6.b.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-6.b.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.i.1.4 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.0-12.i.1.7 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.48.1-12.l.1.8 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.48.1-12.l.1.11 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.192.3-12.p.1.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.192.3-12.q.1.3 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.288.5-12.bd.1.2 | $12$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.192.3-24.ha.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hb.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hi.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hj.1.7 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ho.1.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.hp.1.11 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.5-24.bu.1.13 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.bw.1.9 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.fw.1.9 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.fy.1.13 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 36.288.5-36.o.1.4 | $36$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 36.288.9-36.ch.1.5 | $36$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 36.288.9-36.ci.1.4 | $36$ | $3$ | $3$ | $9$ | $2$ | $1^{8}$ |
| 60.192.3-60.bo.1.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.192.3-60.bp.1.6 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.480.17-60.ky.1.9 | $60$ | $5$ | $5$ | $17$ | $7$ | $1^{16}$ |
| 60.576.17-60.gk.1.10 | $60$ | $6$ | $6$ | $17$ | $2$ | $1^{16}$ |
| 60.960.33-60.lo.1.21 | $60$ | $10$ | $10$ | $33$ | $16$ | $1^{32}$ |
| 84.192.3-84.bl.1.4 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.192.3-84.bm.1.2 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tw.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.tx.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ua.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ub.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ug.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.uh.1.17 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.gy.1.17 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.gz.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.me.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.mf.1.17 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.192.3-132.bk.1.2 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.192.3-132.bl.1.5 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bp.1.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bq.1.5 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ri.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rj.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rm.1.4 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rn.1.20 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rs.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.rt.1.20 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.5-168.ii.1.18 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.ij.1.18 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.ne.1.18 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.nf.1.18 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.192.3-204.bo.1.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.192.3-204.bp.1.6 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.bl.1.4 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.192.3-228.bm.1.2 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ri.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rj.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rm.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rn.1.10 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rs.1.14 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.rt.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.5-264.fg.1.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.fh.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.iq.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.ir.1.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.192.3-276.bk.1.2 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.192.3-276.bl.1.5 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tw.1.7 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.tx.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ua.1.3 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ub.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ug.1.15 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.uh.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.fo.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.fp.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.iy.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.iz.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |