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Properties

Label 12.144.7.bl.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 $2^{2}\cdot4^{2}$
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: 12.144.7.39

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&2\\4&1\end{bmatrix}$, $\begin{bmatrix}3&4\\8&3\end{bmatrix}$, $\begin{bmatrix}9&11\\8&3\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2\times \SD_{16}$
Contains $-I$: yes
Quadratic refinements: 12.288.7-12.bl.1.1, 12.288.7-12.bl.1.2, 24.288.7-12.bl.1.1, 24.288.7-12.bl.1.2, 24.288.7-12.bl.1.3, 24.288.7-12.bl.1.4, 24.288.7-12.bl.1.5, 24.288.7-12.bl.1.6, 60.288.7-12.bl.1.1, 60.288.7-12.bl.1.2, 84.288.7-12.bl.1.1, 84.288.7-12.bl.1.2, 120.288.7-12.bl.1.1, 120.288.7-12.bl.1.2, 120.288.7-12.bl.1.3, 120.288.7-12.bl.1.4, 120.288.7-12.bl.1.5, 120.288.7-12.bl.1.6, 132.288.7-12.bl.1.1, 132.288.7-12.bl.1.2, 156.288.7-12.bl.1.1, 156.288.7-12.bl.1.2, 168.288.7-12.bl.1.1, 168.288.7-12.bl.1.2, 168.288.7-12.bl.1.3, 168.288.7-12.bl.1.4, 168.288.7-12.bl.1.5, 168.288.7-12.bl.1.6, 204.288.7-12.bl.1.1, 204.288.7-12.bl.1.2, 228.288.7-12.bl.1.1, 228.288.7-12.bl.1.2, 264.288.7-12.bl.1.1, 264.288.7-12.bl.1.2, 264.288.7-12.bl.1.3, 264.288.7-12.bl.1.4, 264.288.7-12.bl.1.5, 264.288.7-12.bl.1.6, 276.288.7-12.bl.1.1, 276.288.7-12.bl.1.2, 312.288.7-12.bl.1.1, 312.288.7-12.bl.1.2, 312.288.7-12.bl.1.3, 312.288.7-12.bl.1.4, 312.288.7-12.bl.1.5, 312.288.7-12.bl.1.6
Cyclic 12-isogeny field degree: $4$
Cyclic 12-torsion field degree: $16$
Full 12-torsion field degree: $32$

Jacobian

Conductor: $2^{22}\cdot3^{12}$
Simple: no
Squarefree: no
Decomposition: $1^{7}$
Newforms: 24.2.a.a, 36.2.a.a$^{2}$, 48.2.a.a, 72.2.a.a, 144.2.a.a$^{2}$

Models

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

$ 0 $ $=$ $ x v - y u - y v - z u + z v $
$=$ $x u + y u - 2 y v + z v$
$=$ $x t + y w + y t - z w + z t$
$=$ $x w - y w + 2 y t + z t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{12} + 90 x^{10} y^{2} + 30 x^{10} z^{2} + 870 x^{8} y^{4} - 36 x^{8} y^{2} z^{2} - 74 x^{8} z^{4} + \cdots + 144 z^{12} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

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

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.q.1 :

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

Equation of the image curve:

$0$ $=$ $ 6X^{2}-Z^{2}+W^{2} $
$=$ $ 3X^{3}+24Y^{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
12.72.2.q.1 $12$ $2$ $2$ $2$ $0$ $1^{5}$
12.72.2.r.1 $12$ $2$ $2$ $2$ $0$ $1^{5}$
12.72.3.bj.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.3.dq.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.3.dr.1 $12$ $2$ $2$ $3$ $0$ $1^{4}$
12.72.4.q.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.cmu.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.cnc.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.cxg.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.cxn.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.dhb.1 $24$ $2$ $2$ $17$ $2$ $1^{10}$
24.288.17.dhk.1 $24$ $2$ $2$ $17$ $3$ $1^{10}$
24.288.17.djo.1 $24$ $2$ $2$ $17$ $1$ $1^{10}$
24.288.17.djw.1 $24$ $2$ $2$ $17$ $6$ $1^{10}$
36.432.31.bl.1 $36$ $3$ $3$ $31$ $6$ $1^{24}$
36.1296.91.cs.1 $36$ $9$ $9$ $91$ $36$ $1^{48}\cdot2^{18}$
60.720.55.oa.1 $60$ $5$ $5$ $55$ $22$ $1^{48}$
60.864.61.bsx.1 $60$ $6$ $6$ $61$ $10$ $1^{54}$
60.1440.109.cjg.1 $60$ $10$ $10$ $109$ $46$ $1^{102}$
120.288.17.tmm.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.tnc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.trk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.tsa.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wom.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wpc.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wtk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.wua.1 $120$ $2$ $2$ $17$ $?$ not computed
168.288.17.rnu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.rok.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.rss.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.rti.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.txi.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.txy.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.ucg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.ucw.1 $168$ $2$ $2$ $17$ $?$ not computed
264.288.17.rop.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rpf.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rtn.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.rud.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.tyf.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.tyv.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.udd.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.udt.1 $264$ $2$ $2$ $17$ $?$ not computed
312.288.17.rny.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.roo.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.rsw.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.rtm.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.txm.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.tyc.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.uck.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.uda.1 $312$ $2$ $2$ $17$ $?$ not computed