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Properties

Label 12.144.7.bi.1
Level $12$
Index $144$
Genus $7$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $144$
Index: $144$ $\PSL_2$-index:$144$
Genus: $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $12^{12}$ Cusp orbits $2^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12B7
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.144.7.16

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}3&4\\4&3\end{bmatrix}$, $\begin{bmatrix}3&4\\8&3\end{bmatrix}$, $\begin{bmatrix}5&11\\4&11\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2\times \SD_{16}$
Contains $-I$: yes
Quadratic refinements: 12.288.7-12.bi.1.1, 12.288.7-12.bi.1.2, 24.288.7-12.bi.1.1, 24.288.7-12.bi.1.2, 24.288.7-12.bi.1.3, 24.288.7-12.bi.1.4, 24.288.7-12.bi.1.5, 24.288.7-12.bi.1.6, 60.288.7-12.bi.1.1, 60.288.7-12.bi.1.2, 84.288.7-12.bi.1.1, 84.288.7-12.bi.1.2, 120.288.7-12.bi.1.1, 120.288.7-12.bi.1.2, 120.288.7-12.bi.1.3, 120.288.7-12.bi.1.4, 120.288.7-12.bi.1.5, 120.288.7-12.bi.1.6, 132.288.7-12.bi.1.1, 132.288.7-12.bi.1.2, 156.288.7-12.bi.1.1, 156.288.7-12.bi.1.2, 168.288.7-12.bi.1.1, 168.288.7-12.bi.1.2, 168.288.7-12.bi.1.3, 168.288.7-12.bi.1.4, 168.288.7-12.bi.1.5, 168.288.7-12.bi.1.6, 204.288.7-12.bi.1.1, 204.288.7-12.bi.1.2, 228.288.7-12.bi.1.1, 228.288.7-12.bi.1.2, 264.288.7-12.bi.1.1, 264.288.7-12.bi.1.2, 264.288.7-12.bi.1.3, 264.288.7-12.bi.1.4, 264.288.7-12.bi.1.5, 264.288.7-12.bi.1.6, 276.288.7-12.bi.1.1, 276.288.7-12.bi.1.2, 312.288.7-12.bi.1.1, 312.288.7-12.bi.1.2, 312.288.7-12.bi.1.3, 312.288.7-12.bi.1.4, 312.288.7-12.bi.1.5, 312.288.7-12.bi.1.6
Cyclic 12-isogeny field degree: $4$
Cyclic 12-torsion field degree: $16$
Full 12-torsion field degree: $32$

Jacobian

Conductor: $2^{22}\cdot3^{11}$
Simple: no
Squarefree: no
Decomposition: $1^{7}$
Newforms: 24.2.a.a$^{2}$, 36.2.a.a$^{2}$, 48.2.a.a, 144.2.a.a$^{2}$

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $ $=$ $ w v - t u $
$=$ $y u + z v$
$=$ $x v + z u$
$=$ $y w + z t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{8} - 2 x^{6} y^{2} + 2 x^{6} z^{2} + 6 x^{4} y^{4} + 12 x^{4} y^{2} z^{2} + 6 x^{4} z^{4} - 2 x^{2} y^{6} + \cdots + z^{8} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}u$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 12.72.4.p.1 :

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle u-v$
$\displaystyle W$ $=$ $\displaystyle -w-t$

Equation of the image curve:

$0$ $=$ $ 2X^{2}+Z^{2}-W^{2} $
$=$ $ X^{3}+8Y^{3}+XZ^{2} $

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.3.j.1 $12$ $3$ $3$ $3$ $0$ $1^{4}$
12.72.2.g.1 $12$ $2$ $2$ $2$ $0$ $1^{5}$
12.72.2.h.1 $12$ $2$ $2$ $2$ $0$ $1^{5}$
12.72.3.bi.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.3.dy.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.3.dz.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.4.p.1 $12$ $2$ $2$ $4$ $0$ $1^{3}$
12.72.4.r.1 $12$ $2$ $2$ $4$ $0$ $1^{3}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.17.clm.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.clu.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.csi.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.csu.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.dfq.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.dgc.1 $24$ $2$ $2$ $17$ $6$ $1^{10}$
24.288.17.dig.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.dio.1 $24$ $2$ $2$ $17$ $3$ $1^{10}$
36.432.31.bi.1 $36$ $3$ $3$ $31$ $6$ $1^{24}$
36.1296.91.cp.1 $36$ $9$ $9$ $91$ $32$ $1^{48}\cdot2^{18}$
60.720.55.nv.1 $60$ $5$ $5$ $55$ $16$ $1^{48}$
60.864.61.bsq.1 $60$ $6$ $6$ $61$ $5$ $1^{54}$
60.1440.109.cix.1 $60$ $10$ $10$ $109$ $33$ $1^{102}$
120.288.17.tcm.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.tdc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.thk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.tia.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wem.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wfc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wjk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wka.1 $120$ $2$ $2$ $17$ $?$ not computed
168.288.17.rdu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.rek.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.ris.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.rji.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.tni.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.tny.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.tsg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.tsw.1 $168$ $2$ $2$ $17$ $?$ not computed
264.288.17.rep.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rff.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rjn.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rkd.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.tof.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.tov.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.ttd.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.ttt.1 $264$ $2$ $2$ $17$ $?$ not computed
312.288.17.rdy.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.reo.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.riw.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.rjm.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.tnm.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.toc.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.tsk.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.tta.1 $312$ $2$ $2$ $17$ $?$ not computed