Invariants
Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $3$ | ||||||
$\overline{\Q}$-gonality: | $3$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12A4 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.144.4.10 |
Level structure
$\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}1&4\\10&3\end{bmatrix}$, $\begin{bmatrix}3&10\\10&11\end{bmatrix}$, $\begin{bmatrix}11&2\\8&1\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: | $C_2\times \SD_{16}$ |
Contains $-I$: | no $\quad$ (see 12.72.4.e.1 for the level structure with $-I$) |
Cyclic 12-isogeny field degree: | $4$ |
Cyclic 12-torsion field degree: | $8$ |
Full 12-torsion field degree: | $32$ |
Jacobian
Conductor: | $2^{10}\cdot3^{8}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{4}$ |
Newforms: | 36.2.a.a$^{3}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
$ 0 $ | $=$ | $ 3 x^{2} + z^{2} - w^{2} $ |
$=$ | $ - x z w + 12 y^{3}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{5} - x z^{4} - 6 y^{3} z^{2} $ |
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.
Canonical model |
---|
$(0:0:1:1)$, $(0:0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{4}-z^{2}w^{2}+w^{4})^{3}}{w^{4}z^{4}(z-w)^{2}(z+w)^{2}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 12.72.4.e.1 :
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
$\displaystyle Z$ | $=$ | $\displaystyle z-w$ |
Equation of the image curve:
$0$ | $=$ | $ 9X^{5}-6Y^{3}Z^{2}-XZ^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.48.0-12.c.1.1 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
12.72.2-12.b.1.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.72.2-12.b.1.6 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.72.2-12.c.1.6 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.72.2-12.c.1.7 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.72.2-12.d.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.72.2-12.d.1.6 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.288.7-12.l.1.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.288.7-12.n.1.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.288.7-12.s.1.3 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.288.7-12.u.1.3 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
24.288.7-24.bv.1.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
24.288.7-24.ch.1.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
24.288.7-24.dm.1.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
24.288.7-24.dy.1.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
24.288.8-24.y.1.2 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.y.2.2 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.z.1.28 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.z.2.22 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.ba.1.2 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.ba.2.17 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bb.1.12 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bb.2.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bc.1.22 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bc.2.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bd.1.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bd.2.25 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.be.1.26 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.be.2.24 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bf.1.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.8-24.bf.2.4 | $24$ | $2$ | $2$ | $8$ | $0$ | $2^{2}$ |
24.288.9-24.cz.1.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.288.9-24.dp.1.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.288.9-24.ge.1.7 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
24.288.9-24.gg.1.7 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
24.288.9-24.ii.1.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.288.9-24.ik.1.8 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.288.9-24.jc.1.4 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
24.288.9-24.je.1.2 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
36.432.16-36.e.1.1 | $36$ | $3$ | $3$ | $16$ | $3$ | $1^{12}$ |
36.1296.46-36.j.1.3 | $36$ | $9$ | $9$ | $46$ | $13$ | $1^{36}\cdot2^{3}$ |
60.288.7-60.l.1.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
60.288.7-60.n.1.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
60.288.7-60.v.1.5 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
60.288.7-60.x.1.6 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
60.720.28-60.e.1.6 | $60$ | $5$ | $5$ | $28$ | $10$ | $1^{24}$ |
60.864.31-60.e.1.4 | $60$ | $6$ | $6$ | $31$ | $3$ | $1^{27}$ |
60.1440.55-60.bi.1.2 | $60$ | $10$ | $10$ | $55$ | $20$ | $1^{51}$ |
84.288.7-84.l.1.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.288.7-84.n.1.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.288.7-84.s.1.5 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.288.7-84.u.1.7 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.bv.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.ch.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.dl.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.7-120.dx.1.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.288.8-120.y.1.3 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.y.2.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.z.1.58 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.z.2.54 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.ba.1.4 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.ba.2.33 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bb.1.54 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bb.2.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bc.1.58 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bc.2.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bd.1.2 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bd.2.49 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.be.1.58 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.be.2.58 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bf.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.8-120.bf.2.5 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
120.288.9-120.ge.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.gg.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.ho.1.39 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.hq.1.39 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.jw.1.39 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.jy.1.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.ky.1.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.288.9-120.la.1.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
132.288.7-132.l.1.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.288.7-132.n.1.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.288.7-132.s.1.5 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.288.7-132.u.1.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.l.1.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.n.1.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.s.1.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.u.1.6 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.bv.1.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.ch.1.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.de.1.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.dq.1.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.8-168.y.1.1 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.y.2.3 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.z.1.57 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.z.2.53 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.ba.1.4 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.ba.2.33 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bb.1.53 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bb.2.1 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bc.1.53 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bc.2.1 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bd.1.2 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bd.2.49 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.be.1.57 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.be.2.57 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bf.1.1 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.8-168.bf.2.5 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
168.288.9-168.ge.1.5 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.gg.1.5 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.ho.1.39 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.hq.1.39 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.js.1.39 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.ju.1.40 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.ku.1.5 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.288.9-168.kw.1.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
204.288.7-204.l.1.6 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.288.7-204.n.1.6 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.288.7-204.s.1.5 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.288.7-204.u.1.7 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.288.7-228.l.1.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.288.7-228.n.1.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.288.7-228.s.1.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.288.7-228.u.1.7 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.288.7-264.bv.1.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.288.7-264.ch.1.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.288.7-264.de.1.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.288.7-264.dq.1.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.288.8-264.y.1.3 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.y.2.1 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.z.1.57 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.z.2.57 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.ba.1.4 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.ba.2.33 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bb.1.53 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bb.2.1 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bc.1.57 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bc.2.1 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bd.1.2 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bd.2.49 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.be.1.57 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.be.2.57 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bf.1.1 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.8-264.bf.2.5 | $264$ | $2$ | $2$ | $8$ | $?$ | not computed |
264.288.9-264.ge.1.5 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.gg.1.5 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.ho.1.39 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.hq.1.39 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.js.1.39 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.ju.1.40 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.ku.1.5 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.288.9-264.kw.1.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
276.288.7-276.l.1.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.288.7-276.n.1.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.288.7-276.s.1.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.288.7-276.u.1.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-312.bv.1.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-312.ch.1.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-312.de.1.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-312.dq.1.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.8-312.y.1.3 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.y.2.1 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.z.1.58 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.z.2.54 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.ba.1.4 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.ba.2.33 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bb.1.54 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bb.2.1 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bc.1.58 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bc.2.1 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bd.1.2 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bd.2.49 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.be.1.58 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.be.2.58 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bf.1.1 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.8-312.bf.2.5 | $312$ | $2$ | $2$ | $8$ | $?$ | not computed |
312.288.9-312.ge.1.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.gg.1.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.ho.1.39 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.hq.1.39 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.js.1.39 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.ju.1.40 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.ku.1.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.288.9-312.kw.1.2 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |