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Properties

Label 12.72.4.o.1
Level $12$
Index $72$
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: $72$ $\PSL_2$-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: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12C4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.72.4.27

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}7&0\\0&5\end{bmatrix}$, $\begin{bmatrix}7&2\\2&11\end{bmatrix}$, $\begin{bmatrix}7&8\\2&5\end{bmatrix}$, $\begin{bmatrix}9&10\\4&9\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times \SD_{16}$
Contains $-I$: yes
Quadratic refinements: 12.144.4-12.o.1.1, 12.144.4-12.o.1.2, 12.144.4-12.o.1.3, 12.144.4-12.o.1.4, 12.144.4-12.o.1.5, 12.144.4-12.o.1.6, 12.144.4-12.o.1.7, 12.144.4-12.o.1.8, 24.144.4-12.o.1.1, 24.144.4-12.o.1.2, 24.144.4-12.o.1.3, 24.144.4-12.o.1.4, 24.144.4-12.o.1.5, 24.144.4-12.o.1.6, 24.144.4-12.o.1.7, 24.144.4-12.o.1.8, 24.144.4-12.o.1.9, 24.144.4-12.o.1.10, 24.144.4-12.o.1.11, 24.144.4-12.o.1.12, 60.144.4-12.o.1.1, 60.144.4-12.o.1.2, 60.144.4-12.o.1.3, 60.144.4-12.o.1.4, 60.144.4-12.o.1.5, 60.144.4-12.o.1.6, 60.144.4-12.o.1.7, 60.144.4-12.o.1.8, 84.144.4-12.o.1.1, 84.144.4-12.o.1.2, 84.144.4-12.o.1.3, 84.144.4-12.o.1.4, 84.144.4-12.o.1.5, 84.144.4-12.o.1.6, 84.144.4-12.o.1.7, 84.144.4-12.o.1.8, 120.144.4-12.o.1.1, 120.144.4-12.o.1.2, 120.144.4-12.o.1.3, 120.144.4-12.o.1.4, 120.144.4-12.o.1.5, 120.144.4-12.o.1.6, 120.144.4-12.o.1.7, 120.144.4-12.o.1.8, 120.144.4-12.o.1.9, 120.144.4-12.o.1.10, 120.144.4-12.o.1.11, 120.144.4-12.o.1.12, 132.144.4-12.o.1.1, 132.144.4-12.o.1.2, 132.144.4-12.o.1.3, 132.144.4-12.o.1.4, 132.144.4-12.o.1.5, 132.144.4-12.o.1.6, 132.144.4-12.o.1.7, 132.144.4-12.o.1.8, 156.144.4-12.o.1.1, 156.144.4-12.o.1.2, 156.144.4-12.o.1.3, 156.144.4-12.o.1.4, 156.144.4-12.o.1.5, 156.144.4-12.o.1.6, 156.144.4-12.o.1.7, 156.144.4-12.o.1.8, 168.144.4-12.o.1.1, 168.144.4-12.o.1.2, 168.144.4-12.o.1.3, 168.144.4-12.o.1.4, 168.144.4-12.o.1.5, 168.144.4-12.o.1.6, 168.144.4-12.o.1.7, 168.144.4-12.o.1.8, 168.144.4-12.o.1.9, 168.144.4-12.o.1.10, 168.144.4-12.o.1.11, 168.144.4-12.o.1.12, 204.144.4-12.o.1.1, 204.144.4-12.o.1.2, 204.144.4-12.o.1.3, 204.144.4-12.o.1.4, 204.144.4-12.o.1.5, 204.144.4-12.o.1.6, 204.144.4-12.o.1.7, 204.144.4-12.o.1.8, 228.144.4-12.o.1.1, 228.144.4-12.o.1.2, 228.144.4-12.o.1.3, 228.144.4-12.o.1.4, 228.144.4-12.o.1.5, 228.144.4-12.o.1.6, 228.144.4-12.o.1.7, 228.144.4-12.o.1.8, 264.144.4-12.o.1.1, 264.144.4-12.o.1.2, 264.144.4-12.o.1.3, 264.144.4-12.o.1.4, 264.144.4-12.o.1.5, 264.144.4-12.o.1.6, 264.144.4-12.o.1.7, 264.144.4-12.o.1.8, 264.144.4-12.o.1.9, 264.144.4-12.o.1.10, 264.144.4-12.o.1.11, 264.144.4-12.o.1.12, 276.144.4-12.o.1.1, 276.144.4-12.o.1.2, 276.144.4-12.o.1.3, 276.144.4-12.o.1.4, 276.144.4-12.o.1.5, 276.144.4-12.o.1.6, 276.144.4-12.o.1.7, 276.144.4-12.o.1.8, 312.144.4-12.o.1.1, 312.144.4-12.o.1.2, 312.144.4-12.o.1.3, 312.144.4-12.o.1.4, 312.144.4-12.o.1.5, 312.144.4-12.o.1.6, 312.144.4-12.o.1.7, 312.144.4-12.o.1.8, 312.144.4-12.o.1.9, 312.144.4-12.o.1.10, 312.144.4-12.o.1.11, 312.144.4-12.o.1.12
Cyclic 12-isogeny field degree: $4$
Cyclic 12-torsion field degree: $16$
Full 12-torsion field degree: $64$

Jacobian

Conductor: $2^{11}\cdot3^{7}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 24.2.a.a, 36.2.a.a$^{2}$, 144.2.a.b

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 24 x^{2} - 3 y^{2} + z^{2} - w^{2} $
$=$ $3 x y^{2} + 3 x z^{2} + x w^{2} + 2 y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 36 x^{6} - 12 x^{4} z^{2} + 24 x^{2} y^{2} z^{2} - x^{2} z^{4} - 12 y^{4} z^{2} + y^{2} z^{4} $
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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 between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^7\,\frac{816xyz^{9}w+5832xyz^{7}w^{3}+10080xyz^{5}w^{5}+4656xyz^{3}w^{7}+360xyzw^{9}+39y^{2}z^{10}+870y^{2}z^{8}w^{2}+3069y^{2}z^{6}w^{4}+2988y^{2}z^{4}w^{6}+705y^{2}z^{2}w^{8}+18y^{2}w^{10}-15z^{12}-167z^{10}w^{2}-273z^{8}w^{4}+121z^{6}w^{6}+253z^{4}w^{8}+75z^{2}w^{10}+4w^{12}}{z^{4}(60xyz^{5}w+216xyz^{3}w^{3}+72xyzw^{5}+3y^{2}z^{6}+51y^{2}z^{4}w^{2}+63y^{2}z^{2}w^{4}+6y^{2}w^{6}-z^{8}-8z^{6}w^{2}+3z^{4}w^{4}+5z^{2}w^{6}+w^{8})}$

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
12.36.2.d.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$
12.36.2.e.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$
12.36.2.h.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
12.144.7.k.1 $12$ $2$ $2$ $7$ $0$ $1^{3}$
12.144.7.n.1 $12$ $2$ $2$ $7$ $0$ $1^{3}$
12.144.7.s.1 $12$ $2$ $2$ $7$ $0$ $1^{3}$
12.144.7.t.1 $12$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.bu.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.cn.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.ds.1 $24$ $2$ $2$ $7$ $2$ $1^{3}$
24.144.7.dx.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.8.ea.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.ea.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.eb.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.eb.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
36.216.16.o.1 $36$ $3$ $3$ $16$ $3$ $1^{12}$
36.648.46.t.1 $36$ $9$ $9$ $46$ $13$ $1^{24}\cdot2^{9}$
60.144.7.dp.1 $60$ $2$ $2$ $7$ $0$ $1^{3}$
60.144.7.ds.1 $60$ $2$ $2$ $7$ $1$ $1^{3}$
60.144.7.ed.1 $60$ $2$ $2$ $7$ $1$ $1^{3}$
60.144.7.eg.1 $60$ $2$ $2$ $7$ $0$ $1^{3}$
60.360.28.bi.1 $60$ $5$ $5$ $28$ $9$ $1^{24}$
60.432.31.ck.1 $60$ $6$ $6$ $31$ $3$ $1^{27}$
60.720.55.ie.1 $60$ $10$ $10$ $55$ $18$ $1^{51}$
84.144.7.df.1 $84$ $2$ $2$ $7$ $?$ not computed
84.144.7.di.1 $84$ $2$ $2$ $7$ $?$ not computed
84.144.7.dt.1 $84$ $2$ $2$ $7$ $?$ not computed
84.144.7.dw.1 $84$ $2$ $2$ $7$ $?$ not computed
120.144.7.tx.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.us.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.xj.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.ye.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.8.ls.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.ls.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.lt.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.lt.2 $120$ $2$ $2$ $8$ $?$ not computed
132.144.7.df.1 $132$ $2$ $2$ $7$ $?$ not computed
132.144.7.di.1 $132$ $2$ $2$ $7$ $?$ not computed
132.144.7.dt.1 $132$ $2$ $2$ $7$ $?$ not computed
132.144.7.dw.1 $132$ $2$ $2$ $7$ $?$ not computed
156.144.7.df.1 $156$ $2$ $2$ $7$ $?$ not computed
156.144.7.di.1 $156$ $2$ $2$ $7$ $?$ not computed
156.144.7.dt.1 $156$ $2$ $2$ $7$ $?$ not computed
156.144.7.dw.1 $156$ $2$ $2$ $7$ $?$ not computed
168.144.7.sx.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.ts.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.wj.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.xe.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.8.lk.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.lk.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ll.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ll.2 $168$ $2$ $2$ $8$ $?$ not computed
204.144.7.df.1 $204$ $2$ $2$ $7$ $?$ not computed
204.144.7.di.1 $204$ $2$ $2$ $7$ $?$ not computed
204.144.7.dt.1 $204$ $2$ $2$ $7$ $?$ not computed
204.144.7.dw.1 $204$ $2$ $2$ $7$ $?$ not computed
228.144.7.df.1 $228$ $2$ $2$ $7$ $?$ not computed
228.144.7.di.1 $228$ $2$ $2$ $7$ $?$ not computed
228.144.7.dt.1 $228$ $2$ $2$ $7$ $?$ not computed
228.144.7.dw.1 $228$ $2$ $2$ $7$ $?$ not computed
264.144.7.sx.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.ts.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.wj.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.xe.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.8.ll.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.ll.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.lm.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.lm.2 $264$ $2$ $2$ $8$ $?$ not computed
276.144.7.df.1 $276$ $2$ $2$ $7$ $?$ not computed
276.144.7.di.1 $276$ $2$ $2$ $7$ $?$ not computed
276.144.7.dt.1 $276$ $2$ $2$ $7$ $?$ not computed
276.144.7.dw.1 $276$ $2$ $2$ $7$ $?$ not computed
312.144.7.sx.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.ts.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.wj.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.xe.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.8.ls.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.ls.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.lt.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.lt.2 $312$ $2$ $2$ $8$ $?$ not computed