Properties

Label 12.144.4-12.e.1.1
Level $12$
Index $144$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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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}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{5} - x z^{4} - 6 y^{3} z^{2} $
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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