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Properties

Label 12.144.3-12.o.1.5
Level $12$
Index $144$
Genus $3$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $72$
Index: $144$ $\PSL_2$-index:$72$
Genus: $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $6^{4}\cdot12^{4}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12G3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.144.3.61

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&0\\6&11\end{bmatrix}$, $\begin{bmatrix}3&8\\2&3\end{bmatrix}$, $\begin{bmatrix}11&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&6\\0&5\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: no $\quad$ (see 12.72.3.o.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: $2^{7}\cdot3^{6}$
Simple: no
Squarefree: no
Decomposition: $1^{3}$
Newforms: 36.2.a.a$^{2}$, 72.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ x y z - x y t - y^{2} z - y z w $
$=$ $x z^{2} - x z t - y z^{2} - z^{2} w$
$=$ $x z t - x t^{2} - y z t - z w t$
$=$ $x^{2} z - x^{2} t - x y z - x z w$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} y + x^{4} z + 3 x^{2} y^{2} z + 18 x^{2} y z^{2} + 6 x^{2} z^{3} + 9 y z^{4} + 9 z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} + \left(x^{4} + 1\right) y $ $=$ $ 6x^{6} + 67x^{4} + 54x^{2} + 20 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:1:0:0)$, $(0:-1:0:1:0)$, $(1:1:0:0:0)$, $(1/2:1:0:0:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4}\cdot\frac{846207xy^{8}t^{2}-80771958xy^{6}t^{4}+884635920xy^{4}t^{6}-1247156184xy^{2}t^{8}-1584325698xt^{10}+1296y^{9}z^{2}+34992y^{9}zt-386127y^{9}t^{2}-1028187y^{7}z^{2}t^{2}-10785015y^{7}zt^{3}+31138047y^{7}t^{4}+40023009y^{5}z^{2}t^{4}+287253252y^{5}zt^{5}-218391165y^{5}t^{6}-283898862y^{3}z^{2}t^{6}-1187292897y^{3}zt^{7}-249062559y^{3}t^{8}+789180171yz^{2}t^{8}+355441494yzt^{9}+1120989536yt^{10}+256z^{10}w+4864z^{9}wt+37632z^{8}wt^{2}+151808z^{7}wt^{3}+338432z^{6}wt^{4}+408336z^{5}wt^{5}+453503z^{4}wt^{6}+2147580z^{3}wt^{7}+99079511z^{2}wt^{8}+6382639342zwt^{9}+314928w^{11}+105057w^{9}t^{2}+43794000w^{7}t^{4}-644485365w^{5}t^{6}+2526299658w^{3}t^{8}+2895179378wt^{10}}{t^{4}(1710xy^{4}t^{2}-17334xy^{2}t^{4}-1854xt^{6}+9y^{5}z^{2}+135y^{5}zt-729y^{5}t^{2}-927y^{3}z^{2}t^{2}-6915y^{3}zt^{3}+3351y^{3}t^{4}+4697yz^{2}t^{4}+12260yzt^{5}+9866yt^{6}+16z^{4}wt^{2}+16z^{3}wt^{3}+287z^{2}wt^{4}+31828zwt^{5}-855w^{5}t^{2}+11736w^{3}t^{4}+12211wt^{6})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.72.3.o.1 :

$\displaystyle X$ $=$ $\displaystyle w$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}t$

Equation of the image curve:

$0$ $=$ $ X^{4}Y+X^{4}Z+3X^{2}Y^{2}Z+18X^{2}YZ^{2}+6X^{2}Z^{3}+9YZ^{4}+9Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.72.3.o.1 :

$\displaystyle X$ $=$ $\displaystyle -t$
$\displaystyle Y$ $=$ $\displaystyle -3zw^{2}t-5w^{4}-9w^{2}t^{2}-t^{4}$
$\displaystyle Z$ $=$ $\displaystyle w$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.72.1-6.b.1.1 $12$ $2$ $2$ $1$ $0$ $1^{2}$
12.72.1-6.b.1.4 $12$ $2$ $2$ $1$ $0$ $1^{2}$
12.72.2-12.d.1.3 $12$ $2$ $2$ $2$ $0$ $1$
12.72.2-12.d.1.4 $12$ $2$ $2$ $2$ $0$ $1$
12.72.2-12.g.1.3 $12$ $2$ $2$ $2$ $0$ $1$
12.72.2-12.g.1.4 $12$ $2$ $2$ $2$ $0$ $1$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
12.288.5-12.e.1.1 $12$ $2$ $2$ $5$ $0$ $1^{2}$
12.288.5-12.h.1.1 $12$ $2$ $2$ $5$ $0$ $1^{2}$
12.288.7-12.r.1.1 $12$ $2$ $2$ $7$ $0$ $1^{4}$
12.288.7-12.r.1.2 $12$ $2$ $2$ $7$ $0$ $1^{4}$
12.288.7-12.s.1.3 $12$ $2$ $2$ $7$ $0$ $1^{4}$
12.288.7-12.s.1.5 $12$ $2$ $2$ $7$ $0$ $1^{4}$
12.288.7-12.z.1.3 $12$ $2$ $2$ $7$ $0$ $1^{4}$
12.288.7-12.z.1.5 $12$ $2$ $2$ $7$ $0$ $1^{4}$
24.288.5-24.n.1.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.288.5-24.w.1.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.288.7-24.dj.1.5 $24$ $2$ $2$ $7$ $2$ $1^{4}$
24.288.7-24.dj.1.9 $24$ $2$ $2$ $7$ $2$ $1^{4}$
24.288.7-24.dp.1.7 $24$ $2$ $2$ $7$ $1$ $1^{4}$
24.288.7-24.dp.1.12 $24$ $2$ $2$ $7$ $1$ $1^{4}$
24.288.7-24.hs.1.6 $24$ $2$ $2$ $7$ $1$ $1^{4}$
24.288.7-24.hs.1.16 $24$ $2$ $2$ $7$ $1$ $1^{4}$
36.432.11-36.d.1.7 $36$ $3$ $3$ $11$ $2$ $1^{6}\cdot2$
36.432.15-36.t.1.2 $36$ $3$ $3$ $15$ $4$ $1^{12}$
60.288.5-60.n.1.1 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.288.5-60.p.1.1 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.288.7-60.co.1.5 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.288.7-60.co.1.9 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.288.7-60.cp.1.7 $60$ $2$ $2$ $7$ $0$ $1^{4}$
60.288.7-60.cp.1.12 $60$ $2$ $2$ $7$ $0$ $1^{4}$
60.288.7-60.fc.1.6 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.288.7-60.fc.1.12 $60$ $2$ $2$ $7$ $2$ $1^{4}$
60.720.27-60.o.1.5 $60$ $5$ $5$ $27$ $7$ $1^{24}$
60.864.29-60.da.1.21 $60$ $6$ $6$ $29$ $2$ $1^{26}$
60.1440.53-60.ha.1.3 $60$ $10$ $10$ $53$ $15$ $1^{50}$
84.288.5-84.n.1.1 $84$ $2$ $2$ $5$ $?$ not computed
84.288.5-84.p.1.1 $84$ $2$ $2$ $5$ $?$ not computed
84.288.7-84.ce.1.7 $84$ $2$ $2$ $7$ $?$ not computed
84.288.7-84.ce.1.11 $84$ $2$ $2$ $7$ $?$ not computed
84.288.7-84.cf.1.1 $84$ $2$ $2$ $7$ $?$ not computed
84.288.7-84.cf.1.3 $84$ $2$ $2$ $7$ $?$ not computed
84.288.7-84.ec.1.11 $84$ $2$ $2$ $7$ $?$ not computed
84.288.7-84.ec.1.15 $84$ $2$ $2$ $7$ $?$ not computed
120.288.5-120.bo.1.1 $120$ $2$ $2$ $5$ $?$ not computed
120.288.5-120.bu.1.1 $120$ $2$ $2$ $5$ $?$ not computed
120.288.7-120.nl.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.nl.1.22 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ns.1.5 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.ns.1.21 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bjq.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.288.7-120.bjq.1.23 $120$ $2$ $2$ $7$ $?$ not computed
132.288.5-132.n.1.1 $132$ $2$ $2$ $5$ $?$ not computed
132.288.5-132.p.1.1 $132$ $2$ $2$ $5$ $?$ not computed
132.288.7-132.ce.1.5 $132$ $2$ $2$ $7$ $?$ not computed
132.288.7-132.ce.1.11 $132$ $2$ $2$ $7$ $?$ not computed
132.288.7-132.cf.1.1 $132$ $2$ $2$ $7$ $?$ not computed
132.288.7-132.cf.1.3 $132$ $2$ $2$ $7$ $?$ not computed
132.288.7-132.ec.1.11 $132$ $2$ $2$ $7$ $?$ not computed
132.288.7-132.ec.1.15 $132$ $2$ $2$ $7$ $?$ not computed
156.288.5-156.n.1.1 $156$ $2$ $2$ $5$ $?$ not computed
156.288.5-156.p.1.1 $156$ $2$ $2$ $5$ $?$ not computed
156.288.7-156.ce.1.5 $156$ $2$ $2$ $7$ $?$ not computed
156.288.7-156.ce.1.9 $156$ $2$ $2$ $7$ $?$ not computed
156.288.7-156.cf.1.5 $156$ $2$ $2$ $7$ $?$ not computed
156.288.7-156.cf.1.12 $156$ $2$ $2$ $7$ $?$ not computed
156.288.7-156.ec.1.6 $156$ $2$ $2$ $7$ $?$ not computed
156.288.7-156.ec.1.12 $156$ $2$ $2$ $7$ $?$ not computed
168.288.5-168.bo.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.288.5-168.bu.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.288.7-168.ml.1.8 $168$ $2$ $2$ $7$ $?$ not computed
168.288.7-168.ml.1.22 $168$ $2$ $2$ $7$ $?$ not computed
168.288.7-168.ms.1.4 $168$ $2$ $2$ $7$ $?$ not computed
168.288.7-168.ms.1.11 $168$ $2$ $2$ $7$ $?$ not computed
168.288.7-168.bgk.1.22 $168$ $2$ $2$ $7$ $?$ not computed
168.288.7-168.bgk.1.28 $168$ $2$ $2$ $7$ $?$ not computed
204.288.5-204.n.1.1 $204$ $2$ $2$ $5$ $?$ not computed
204.288.5-204.p.1.1 $204$ $2$ $2$ $5$ $?$ not computed
204.288.7-204.ce.1.5 $204$ $2$ $2$ $7$ $?$ not computed
204.288.7-204.ce.1.9 $204$ $2$ $2$ $7$ $?$ not computed
204.288.7-204.cf.1.7 $204$ $2$ $2$ $7$ $?$ not computed
204.288.7-204.cf.1.12 $204$ $2$ $2$ $7$ $?$ not computed
204.288.7-204.ec.1.6 $204$ $2$ $2$ $7$ $?$ not computed
204.288.7-204.ec.1.12 $204$ $2$ $2$ $7$ $?$ not computed
228.288.5-228.n.1.1 $228$ $2$ $2$ $5$ $?$ not computed
228.288.5-228.p.1.1 $228$ $2$ $2$ $5$ $?$ not computed
228.288.7-228.ce.1.9 $228$ $2$ $2$ $7$ $?$ not computed
228.288.7-228.ce.1.12 $228$ $2$ $2$ $7$ $?$ not computed
228.288.7-228.cf.1.1 $228$ $2$ $2$ $7$ $?$ not computed
228.288.7-228.cf.1.2 $228$ $2$ $2$ $7$ $?$ not computed
228.288.7-228.ec.1.11 $228$ $2$ $2$ $7$ $?$ not computed
228.288.7-228.ec.1.15 $228$ $2$ $2$ $7$ $?$ not computed
264.288.5-264.bo.1.1 $264$ $2$ $2$ $5$ $?$ not computed
264.288.5-264.bu.1.1 $264$ $2$ $2$ $5$ $?$ not computed
264.288.7-264.ml.1.8 $264$ $2$ $2$ $7$ $?$ not computed
264.288.7-264.ml.1.22 $264$ $2$ $2$ $7$ $?$ not computed
264.288.7-264.ms.1.4 $264$ $2$ $2$ $7$ $?$ not computed
264.288.7-264.ms.1.11 $264$ $2$ $2$ $7$ $?$ not computed
264.288.7-264.bgk.1.20 $264$ $2$ $2$ $7$ $?$ not computed
264.288.7-264.bgk.1.30 $264$ $2$ $2$ $7$ $?$ not computed
276.288.5-276.n.1.1 $276$ $2$ $2$ $5$ $?$ not computed
276.288.5-276.p.1.1 $276$ $2$ $2$ $5$ $?$ not computed
276.288.7-276.ce.1.9 $276$ $2$ $2$ $7$ $?$ not computed
276.288.7-276.ce.1.12 $276$ $2$ $2$ $7$ $?$ not computed
276.288.7-276.cf.1.1 $276$ $2$ $2$ $7$ $?$ not computed
276.288.7-276.cf.1.2 $276$ $2$ $2$ $7$ $?$ not computed
276.288.7-276.ec.1.13 $276$ $2$ $2$ $7$ $?$ not computed
276.288.7-276.ec.1.15 $276$ $2$ $2$ $7$ $?$ not computed
312.288.5-312.bo.1.1 $312$ $2$ $2$ $5$ $?$ not computed
312.288.5-312.bu.1.1 $312$ $2$ $2$ $5$ $?$ not computed
312.288.7-312.ml.1.7 $312$ $2$ $2$ $7$ $?$ not computed
312.288.7-312.ml.1.23 $312$ $2$ $2$ $7$ $?$ not computed
312.288.7-312.ms.1.3 $312$ $2$ $2$ $7$ $?$ not computed
312.288.7-312.ms.1.21 $312$ $2$ $2$ $7$ $?$ not computed
312.288.7-312.bgk.1.15 $312$ $2$ $2$ $7$ $?$ not computed
312.288.7-312.bgk.1.21 $312$ $2$ $2$ $7$ $?$ not computed