Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| Elliptic points: | $0$ 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: | 12L3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.3.117 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&4\\6&7\end{bmatrix}$, $\begin{bmatrix}5&6\\0&5\end{bmatrix}$, $\begin{bmatrix}7&0\\6&5\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.96.3.f.1 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $24$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 48.2.a.a, 48.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} w - y z w + z^{3} + z w^{2} $ |
| $=$ | $x^{2} w + y z w + y w^{2} + z^{3} + z^{2} w - z w t$ | |
| $=$ | $2 y z t + y w t + z^{2} t - z w t - z t^{2}$ | |
| $=$ | $2 x^{2} w + y z^{2} - z^{3} - z^{2} w + z w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x^{4} z + 2 x^{3} z^{2} + x^{2} y^{2} z + x^{2} z^{3} + 4 x y^{2} z^{2} + x z^{4} + y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{7} - 5x^{6} + 7x^{5} - 10x^{4} + 7x^{3} - 5x^{2} + x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(1:0:0:0:0)$, $(0:0:0:1/2:1)$ |
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{1}{3}\cdot\frac{1406949348594627936x^{14}+301489146127420272x^{12}t^{2}+442418474880859680x^{10}t^{4}+51750310782745008x^{8}t^{6}+44513492197914432x^{6}t^{8}+1256163847302672x^{4}t^{10}+1619311160831616x^{2}t^{12}+41922967442572191051yw^{13}+31273903730315760822yw^{12}t+94016428739901814086yw^{11}t^{2}+12510493047299454300yw^{10}t^{3}+55253085518351366889yw^{9}t^{4}-20729530327810371198yw^{8}t^{5}+18410525999222671972yw^{7}t^{6}-9118658214926256312yw^{6}t^{7}-429068791271424675yw^{5}t^{8}+4021053853412741274yw^{4}t^{9}-5785727453746370730yw^{3}t^{10}+3506145134046619356yw^{2}t^{11}-1068784399957506993ywt^{12}+127016260637015022yt^{13}-5619798098515363275zw^{13}-25963686287100866043zw^{12}t-22508345921122835334zw^{11}t^{2}-39460982693449689318zw^{10}t^{3}-3073640550212229033zw^{9}t^{4}-15019363413244051273zw^{8}t^{5}+5487500623497676444zw^{7}t^{6}-11495808396855008676zw^{6}t^{7}+43564718531438211zw^{5}t^{8}-1938569214395995917zw^{4}t^{9}-3173017709741137974zw^{3}t^{10}+2515867057076068170zw^{2}t^{11}-998759381299024911zwt^{12}+131183071946744625zt^{13}-20916391529328249564w^{14}-17261074219272140232w^{13}t-37651795925615704809w^{12}t^{2}-4281958422308438448w^{11}t^{3}-5096943670645587414w^{10}t^{4}+10656219472948304776w^{9}t^{5}+2309330512549522293w^{8}t^{6}+162720467936983424w^{7}t^{7}+3552048105227242800w^{6}t^{8}-4313843984961564792w^{5}t^{9}+2603058321766923969w^{4}t^{10}+4515588742001040w^{3}t^{11}-1105189922375331006w^{2}t^{12}+508247494740888216wt^{13}-73278611905970205t^{14}}{195328244980512x^{10}t^{4}+23633009232336x^{8}t^{6}-1608794544672x^{6}t^{8}-33410971440x^{4}t^{10}+12625385472x^{2}t^{12}+2910046852712556yw^{13}-9067217204087604yw^{12}t+9168326626720959yw^{11}t^{2}-3830723039550690yw^{10}t^{3}-86346020235402yw^{9}t^{4}+2277943252712208yw^{8}t^{5}-2138552319539901yw^{7}t^{6}+538924789435710yw^{6}t^{7}+73055099415856yw^{5}t^{8}+18585833513280yw^{4}t^{9}+1586845384560yw^{3}t^{10}-89635201920yw^{2}t^{11}+3568611024ywt^{12}-361293040900350zw^{13}-607565063617902zw^{12}t+2562096936589407zw^{11}t^{2}-4840558935142653zw^{10}t^{3}+2537172662137020zw^{9}t^{4}+1443994889469204zw^{8}t^{5}-1419774613898241zw^{7}t^{6}+425900579226643zw^{6}t^{7}-84022423527184zw^{5}t^{8}-35969976281008zw^{4}t^{9}-6184515787344zw^{3}t^{10}-226323403152zw^{2}t^{11}+71319605904zwt^{12}-7227386640zt^{13}-1583235841530780w^{14}+3915605408069106w^{13}t-3348184850635626w^{12}t^{2}+572112163487394w^{11}t^{3}+2755094483654073w^{10}t^{4}-2711286329434194w^{9}t^{5}+129591544831692w^{8}t^{6}+705857698442654w^{7}t^{7}-268937851942443w^{6}t^{8}+12536104971200w^{5}t^{9}+7068899877952w^{4}t^{10}+942643593312w^{3}t^{11}-29449024320w^{2}t^{12}-1829387808wt^{13}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.96.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}+X^{4}Z+X^{2}Y^{2}Z+2X^{3}Z^{2}+4XY^{2}Z^{2}+X^{2}Z^{3}+Y^{2}Z^{3}+XZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.96.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -xz^{2}w-4xzw^{2}-xw^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.4 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.a.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 12.96.1-12.a.1.4 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 12.96.2-12.a.2.2 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.384.5-12.a.2.2 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.b.2.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.c.1.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.d.1.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.576.13-12.e.1.3 | $12$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 24.384.5-24.c.2.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.g.2.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.bn.2.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.bs.2.2 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 24.384.9-24.t.2.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 24.384.9-24.z.2.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.df.2.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 24.384.9-24.dl.2.2 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 24.384.9-24.fq.2.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot4$ |
| 24.384.9-24.ft.2.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 24.384.9-24.fy.2.2 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 24.384.9-24.gb.2.2 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 36.576.13-36.f.2.3 | $36$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 36.576.19-36.k.2.3 | $36$ | $3$ | $3$ | $19$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 36.576.19-36.o.2.4 | $36$ | $3$ | $3$ | $19$ | $2$ | $1^{8}\cdot4^{2}$ |
| 60.384.5-60.p.1.6 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.384.5-60.q.2.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.384.5-60.s.2.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.384.5-60.t.2.5 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.960.35-60.z.1.3 | $60$ | $5$ | $5$ | $35$ | $3$ | $1^{16}\cdot2^{4}\cdot8$ |
| 60.1152.37-60.ct.2.13 | $60$ | $6$ | $6$ | $37$ | $1$ | $1^{16}\cdot2\cdot4^{2}\cdot8$ |
| 60.1920.69-60.fh.2.2 | $60$ | $10$ | $10$ | $69$ | $7$ | $1^{32}\cdot2^{5}\cdot4^{2}\cdot8^{2}$ |
| 84.384.5-84.p.2.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.q.2.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.s.1.5 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.t.1.3 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.oc.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.oh.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ot.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.oy.2.21 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.9-120.le.2.29 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.lg.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.mg.2.29 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.mi.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.qw.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.qy.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.rm.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.ro.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.384.5-132.p.1.4 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.q.2.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.s.2.4 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.t.2.5 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.p.2.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.q.2.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.s.1.5 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.t.1.5 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.oc.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.oh.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ot.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.oy.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.9-168.le.2.9 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.lg.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.mg.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.mi.2.9 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.qw.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.qy.2.9 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.rm.2.9 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.ro.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.384.5-204.p.1.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.q.2.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.s.2.4 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.t.2.5 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.p.2.6 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.q.2.2 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.s.1.3 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.t.1.3 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.oc.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.oh.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ot.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.oy.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.9-264.le.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.lg.2.7 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.mg.2.14 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.mi.2.7 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.qw.2.8 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.qy.1.3 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.rm.2.8 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.ro.2.4 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.384.5-276.p.1.4 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.q.2.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.s.2.4 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.t.2.5 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.oc.2.24 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.oh.2.24 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ot.2.24 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.oy.2.24 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.9-312.le.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.lg.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.mg.2.26 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.mi.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.qw.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.qy.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.rm.2.30 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.ro.2.26 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |