Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | Cusp orbits | $2^{8}$ | ||
| 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 12V1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.1.53 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&2\\6&5\end{bmatrix}$, $\begin{bmatrix}11&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&4\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.96.1.d.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: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - 2 x y - 2 x w - z^{2} $ |
| $=$ | $x^{2} + 4 x y - 2 y^{2} - 2 y w + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{4} - 2 x^{3} y + 2 x^{2} y^{2} + 4 x^{2} z^{2} - 2 x y z^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
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}{5^{12}}\cdot\frac{9500427440130240000000000xz^{22}w+71622187271342560000000000xz^{20}w^{3}+213052326226751251200000000xz^{18}w^{5}+358613069483897292960000000xz^{16}w^{7}+398176331376854164160000000xz^{14}w^{9}+316164740336368349721600000xz^{12}w^{11}+186789879313780547145600000xz^{10}w^{13}+83180933337059155709088000xz^{8}w^{15}+27749332444780501507958400xz^{6}w^{17}+6708336553281436557048000xz^{4}w^{19}+1081083195221774903618816xz^{2}w^{21}+94485896949912808435368xw^{23}-273511449873600000000000y^{2}z^{22}-4641279602417280000000000y^{2}z^{20}w^{2}-19225600419933160000000000y^{2}z^{18}w^{4}-39168259202668881600000000y^{2}z^{16}w^{6}-49256559387609753600000000y^{2}z^{14}w^{8}-42793476523714012784000000y^{2}z^{12}w^{10}-27165014885819730427200000y^{2}z^{10}w^{12}-12854296973445777979200000y^{2}z^{8}w^{14}-4526450169012903339024000y^{2}z^{6}w^{16}-1153506622787139614232000y^{2}z^{4}w^{18}-195580619433933152743440y^{2}z^{2}w^{20}-18392089209931588070192y^{2}w^{22}-5297236619812320000000000yz^{22}w-45459091036289200000000000yz^{20}w^{3}-147328198712136805600000000yz^{18}w^{5}-263580213635205828480000000yz^{16}w^{7}-306080917892448041120000000yz^{14}w^{9}-251692930961309800428800000yz^{12}w^{11}-153161398250548543749600000yz^{10}w^{13}-70013439382746373062144000yz^{8}w^{15}-23928466641134406843739200yz^{6}w^{17}-5927955081152034057021600yz^{4}w^{19}-978826802303903261917688yz^{2}w^{21}-88739493689912056820192yw^{23}-1094114158869400000000000z^{24}-23771321264940240000000000z^{22}w^{2}-121076215897697720000000000z^{20}w^{4}-299401263409002702000000000z^{18}w^{6}-452319932306855527380000000z^{16}w^{8}-467841203840777234416000000z^{14}w^{10}-352846321985437237829600000z^{12}w^{12}-200171788619924264151600000z^{10}w^{14}-86185556302018861057890000z^{8}w^{16}-27910358149303194482367200z^{6}w^{18}-6562159620277867373045760z^{4}w^{20}-1029388131967466991100876z^{2}w^{22}-86900297004989039949279w^{24}}{z^{4}(3683123200000xz^{18}w+126354391040000xz^{16}w^{3}+1174920916377600xz^{14}w^{5}+4732534117386240xz^{12}w^{7}+10077557645044224xz^{10}w^{9}+12502559771390592xz^{8}w^{11}+9390630625000000xz^{6}w^{13}+4222721093750000xz^{4}w^{15}+1049406328125000xz^{2}w^{17}+111145429687500xw^{19}-39321600000y^{2}z^{18}-4195123200000y^{2}z^{16}w^{2}-66337456640000y^{2}z^{14}w^{4}-376171186713600y^{2}z^{12}w^{6}-1023568025898240y^{2}z^{10}w^{8}-1530556836602048y^{2}z^{8}w^{10}-1333723437500000y^{2}z^{6}w^{12}-677770312500000y^{2}z^{4}w^{14}-186795703125000y^{2}z^{2}w^{16}-21634960937500y^{2}w^{18}-1920204800000yz^{18}w-71716536320000yz^{16}w^{3}-724877236428800yz^{14}w^{5}-3158635633182720yz^{12}w^{7}-7227829581303552yz^{10}w^{9}-9569002461602048yz^{8}w^{11}-7620875156250000yz^{6}w^{13}-3613730000000000yz^{4}w^{15}-942640664062500yz^{2}w^{17}-104385898437500yw^{19}-157286400000z^{20}-18687590400000z^{18}w^{2}-335801044480000z^{16}w^{4}-2210906376243200z^{14}w^{6}-7157726459086080z^{12}w^{8}-13120664717851904z^{10}w^{10}-14603705175448976z^{8}w^{12}-10105645390625000z^{6}w^{14}-4262701162109375z^{4}w^{16}-1006442988281250z^{2}w^{18}-102222402343750w^{20})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.96.1.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 5X^{4}-2X^{3}Y+2X^{2}Y^{2}+4X^{2}Z^{2}-2XYZ^{2}-Z^{4} $ |
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.1.8 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.6 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.d.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 12.96.1-12.d.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.384.5-12.d.1.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.d.2.2 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.e.2.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.384.5-12.e.4.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 12.576.9-12.c.1.4 | $12$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.5-24.br.2.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.br.4.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.1.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.4.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 36.576.9-36.d.1.8 | $36$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 36.576.17-36.d.1.3 | $36$ | $3$ | $3$ | $17$ | $0$ | $1^{8}\cdot4^{2}$ |
| 36.576.17-36.h.2.5 | $36$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot4^{2}$ |
| 60.384.5-60.n.1.6 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.384.5-60.n.2.3 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.384.5-60.o.1.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.384.5-60.o.2.6 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 60.960.33-60.d.2.14 | $60$ | $5$ | $5$ | $33$ | $2$ | $1^{16}\cdot8^{2}$ |
| 60.1152.33-60.d.1.31 | $60$ | $6$ | $6$ | $33$ | $2$ | $1^{16}\cdot8^{2}$ |
| 60.1920.65-60.d.1.21 | $60$ | $10$ | $10$ | $65$ | $6$ | $1^{32}\cdot8^{4}$ |
| 84.384.5-84.n.1.7 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.n.4.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.o.3.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.o.4.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kj.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kj.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kq.1.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kq.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.n.1.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.n.2.3 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.o.1.4 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.o.2.6 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.n.1.7 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.n.4.8 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.o.3.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.o.4.8 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kj.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kj.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kq.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kq.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.n.1.7 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.n.2.3 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.o.1.4 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.o.3.6 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.n.1.7 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.n.4.8 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.o.3.4 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.o.4.8 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kj.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kj.4.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kq.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kq.4.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.n.1.7 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.n.2.3 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.o.1.4 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.o.3.6 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kj.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kj.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kq.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kq.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |