Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| 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: | 12K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.3.31 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}7&3\\6&5\end{bmatrix}$, $\begin{bmatrix}11&1\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_3:D_4$ |
| Contains $-I$: | no $\quad$ (see 12.96.3.p.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $8$ |
| Full 12-torsion field degree: | $24$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 48.2.a.a$^{2}$, 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x u + z u + 2 w^{2} + 2 w t + 2 t^{2} $ |
| $=$ | $x^{2} + 2 x z + z^{2} + 2 w^{2} + 2 w t - t^{2}$ | |
| $=$ | $2 x^{2} - 3 x y - x z + w^{2} + 2 w t$ | |
| $=$ | $x z - 3 y z - 2 z^{2} - w^{2} + t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{6} y^{2} - 32 x^{5} y^{3} + 24 x^{5} y z^{2} + 20 x^{4} y^{4} + 8 x^{4} y^{2} z^{2} + 9 x^{4} z^{4} + \cdots - 27 z^{8} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{7} + 7x^{4} - 8x $ |
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 | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(1:0:1)$, $(-1:0:1)$, $(-1:1:1)$, $(1:-1:1)$ | $(1:0:0)$, $(1:0:1)$, $(0:0:1)$, $(-2:0:1)$ | $(-1/2:-1/6:0:-1/2:1/2:1)$, $(-1/2:-1/6:0:1/2:-1/2:1)$, $(0:1/6:-1/2:0:-1/2:1)$, $(0:1/6:-1/2:0:1/2:1)$ | |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^2}\cdot\frac{9788055552xt^{10}u+14161443840xt^{8}u^{3}+4354864128xt^{6}u^{5}+454495104xt^{4}u^{7}+16148832xt^{2}u^{9}+93308xu^{11}-3260694528yzt^{10}-9917945856yzt^{8}u^{2}-3596120064yzt^{6}u^{4}-379945728yzt^{4}u^{6}-11790144yzt^{2}u^{8}-6552yzu^{10}-8151736320yt^{10}u-8966909952yt^{8}u^{3}-2314635264yt^{6}u^{5}-207753984yt^{4}u^{7}-5895072yt^{2}u^{9}-2173796352z^{2}t^{10}-6611963904z^{2}t^{8}u^{2}-2397413376z^{2}t^{6}u^{4}-253297152z^{2}t^{4}u^{6}-7860096z^{2}t^{2}u^{8}-4368z^{2}u^{10}+3622993920zwt^{9}u+3985293312zwt^{7}u^{3}+1028726784zwt^{5}u^{5}+92335104zwt^{3}u^{7}+2620032zwtu^{9}+730570752zt^{10}u+4198210560zt^{8}u^{3}+1783047168zt^{6}u^{5}+223657344zt^{4}u^{7}+9598752zt^{2}u^{9}+93308zu^{11}+1086898176wt^{11}+1494484992wt^{9}u^{2}-793939968wt^{7}u^{4}-387714816wt^{5}u^{6}-42237504wt^{3}u^{8}-1307832wtu^{10}+5437476864t^{12}+17947256832t^{10}u^{2}+6745731840t^{8}u^{4}+684481536t^{6}u^{6}+14710896t^{4}u^{8}-373992t^{2}u^{10}+u^{12}}{u^{4}(3840xt^{6}u+1976xt^{4}u^{3}+126xt^{2}u^{5}-3xu^{7}-3456yzt^{6}-3312yzt^{4}u^{2}-324yzt^{2}u^{4}-5184yt^{6}u-2160yt^{4}u^{3}-162yt^{2}u^{5}-2304z^{2}t^{6}-2208z^{2}t^{4}u^{2}-216z^{2}t^{2}u^{4}+2304zwt^{5}u+960zwt^{3}u^{3}+72zwtu^{5}-1920zt^{6}u-424zt^{4}u^{3}-54zt^{2}u^{5}-3zu^{7}+1152wt^{7}-48wt^{5}u^{2}-372wt^{3}u^{4}-36wtu^{6}+3456t^{8}+2208t^{6}u^{2}-101t^{4}u^{4}-27t^{2}u^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.96.3.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2t$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{6}Y^{2}-32X^{5}Y^{3}+20X^{4}Y^{4}+8X^{3}Y^{5}-20X^{2}Y^{6}+12XY^{7}-3Y^{8}+24X^{5}YZ^{2}+8X^{4}Y^{2}Z^{2}-76X^{3}Y^{3}Z^{2}+62X^{2}Y^{4}Z^{2}-12XY^{5}Z^{2}-6Y^{6}Z^{2}+9X^{4}Z^{4}+48X^{3}YZ^{4}-114X^{2}Y^{2}Z^{4}+60XY^{3}Z^{4}-21Y^{4}Z^{4}+18X^{2}Z^{6}-72XYZ^{6}+18Y^{2}Z^{6}-27Z^{8} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.96.3.p.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -w^{2}t+w^{2}u-wt^{2}+wtu-t^{3}-\frac{1}{2}t^{2}u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -8zw^{11}-56zw^{10}t+2zw^{10}u-188zw^{9}t^{2}+4zw^{9}tu-408zw^{8}t^{3}-9zw^{8}t^{2}u-624zw^{7}t^{4}-51zw^{7}t^{3}u-696zw^{6}t^{5}-111zw^{6}t^{4}u-564zw^{5}t^{6}-144zw^{5}t^{5}u-312zw^{4}t^{7}-120zw^{4}t^{6}u-96zw^{3}t^{8}-60zw^{3}t^{7}u+8zw^{2}t^{9}-9zw^{2}t^{8}u+20zwt^{10}+7zwt^{9}u+8zt^{11}+5zt^{10}u+2w^{12}+4w^{11}t+4w^{11}u-8w^{10}t^{2}+24w^{10}tu-52w^{9}t^{3}+62w^{9}t^{2}u-120w^{8}t^{4}+90w^{8}t^{3}u-156w^{7}t^{5}+66w^{7}t^{4}u-108w^{6}t^{6}-12w^{6}t^{5}u+12w^{5}t^{7}-84w^{5}t^{6}u+120w^{4}t^{8}-96w^{4}t^{7}u+148w^{3}t^{9}-54w^{3}t^{8}u+104w^{2}t^{10}-10w^{2}t^{9}u+44wt^{11}+6wt^{10}u+10t^{12}+4t^{11}u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w^{3}+2w^{2}t+2wt^{2}+t^{3}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.64.1-12.c.1.4 | $12$ | $3$ | $3$ | $1$ | $0$ | $1^{2}$ |
| 12.96.1-12.o.1.2 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 12.96.1-12.o.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 12.96.2-12.c.1.5 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.96.2-12.c.1.8 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.96.2-12.e.1.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.96.2-12.e.1.6 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.576.13-12.y.1.2 | $12$ | $3$ | $3$ | $13$ | $0$ | $1^{10}$ |
| 24.384.9-24.mg.1.5 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.mq.1.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.nu.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.oa.1.5 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.pd.1.5 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.pj.1.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{6}$ |
| 24.384.9-24.qp.1.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.qz.1.7 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{6}$ |
| 36.576.13-36.s.1.1 | $36$ | $3$ | $3$ | $13$ | $0$ | $1^{10}$ |
| 36.576.19-36.bq.1.1 | $36$ | $3$ | $3$ | $19$ | $3$ | $1^{16}$ |
| 36.576.19-36.br.1.1 | $36$ | $3$ | $3$ | $19$ | $6$ | $1^{16}$ |
| 60.960.35-60.eh.1.3 | $60$ | $5$ | $5$ | $35$ | $14$ | $1^{32}$ |
| 60.1152.37-60.gb.1.10 | $60$ | $6$ | $6$ | $37$ | $5$ | $1^{34}$ |
| 60.1920.69-60.nd.1.11 | $60$ | $10$ | $10$ | $69$ | $31$ | $1^{66}$ |
| 120.384.9-120.cep.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cev.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cfh.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cfj.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cpt.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cpv.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cqh.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.cqn.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cbm.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cbs.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cce.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.ccg.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.clm.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.clo.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cma.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.cmg.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byd.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byj.1.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byv.1.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.byx.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cib.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cid.1.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.cip.1.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.civ.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cdn.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cdt.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cef.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.ceh.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cnt.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.cnv.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.coh.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.con.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |