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Properties

Label 108.216.4-54.a.1.4
Level $108$
Index $216$
Genus $4$
Cusps $12$
$\Q$-cusps $4$

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Invariants

Level: $108$ $\SL_2$-level: $108$ Newform level: $54$
Index: $216$ $\PSL_2$-index:$108$
Genus: $4 = 1 + \frac{ 108 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $4$ are rational) Cusp widths $1^{3}\cdot2^{3}\cdot3^{2}\cdot6^{2}\cdot27\cdot54$ Cusp orbits $1^{4}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 54E4

Level structure

$\GL_2(\Z/108\Z)$-generators: $\begin{bmatrix}36&43\\23&56\end{bmatrix}$, $\begin{bmatrix}80&99\\67&4\end{bmatrix}$, $\begin{bmatrix}85&28\\84&83\end{bmatrix}$, $\begin{bmatrix}98&3\\15&32\end{bmatrix}$
Contains $-I$: no $\quad$ (see 54.108.4.a.1 for the level structure with $-I$)
Cyclic 108-isogeny field degree: $2$
Cyclic 108-torsion field degree: $72$
Full 108-torsion field degree: $139968$

Models

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

$ 0 $ $=$ $ - x w + y z $
$=$ $4 x^{2} y - 2 x y^{2} + y^{3} + z^{3} + z^{2} w + z w^{2}$
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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.

Canonical model
$(0:0:0:1)$, $(-1/4:1/2:-1/2:1)$, $(1:0:0:0)$, $(-1:-1:1:1)$

Maps to other modular curves

$j$-invariant map of degree 108 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{-28991029248x^{18}+114688y^{18}-43486543872x^{15}z^{3}+43486543872x^{15}z^{2}w-25367150592x^{15}zw^{2}-59793997824x^{15}w^{3}+663552xy^{14}w^{3}-2371584y^{15}w^{3}-28538044416x^{12}z^{3}w^{3}-30010245120x^{12}z^{2}w^{4}-1019215872x^{12}zw^{5}-254803968x^{12}w^{6}-46232064xy^{11}w^{6}+9636864y^{12}w^{6}+622854144x^{9}z^{3}w^{6}-2972712960x^{9}z^{2}w^{7}-2505572352x^{9}zw^{8}+873676800x^{9}w^{9}+22930560xy^{8}w^{9}-111814144y^{9}w^{9}+15759360x^{6}z^{3}w^{9}+57397248x^{6}z^{2}w^{10}-252633600x^{6}zw^{11}-258702336x^{6}w^{12}+265705056xy^{5}w^{12}-189352056y^{6}w^{12}+11223360x^{3}z^{3}w^{12}+14668672x^{3}z^{2}w^{13}-7487760x^{3}zw^{14}-69651144x^{3}w^{15}-11990780xy^{2}w^{15}+81188674y^{3}w^{15}+3594050z^{3}w^{15}-3745982z^{2}w^{16}-3745982zw^{17}-7077888w^{18}}{-1024xy^{14}w^{3}+512y^{15}w^{3}+4224xy^{11}w^{6}+384y^{12}w^{6}+884736x^{9}z^{3}w^{6}-1327104x^{9}z^{2}w^{7}-663552x^{9}zw^{8}+1880064x^{9}w^{9}+1344xy^{8}w^{9}-1824y^{9}w^{9}+165888x^{6}z^{3}w^{9}+456192x^{6}z^{2}w^{10}+183168x^{6}zw^{11}+10368x^{6}w^{12}-1216xy^{5}w^{12}+1412y^{6}w^{12}+5184x^{3}z^{3}w^{12}+10912x^{3}z^{2}w^{13}+14344x^{3}zw^{14}+10572x^{3}w^{15}+3834xy^{2}w^{15}-1211y^{3}w^{15}-1211z^{3}w^{15}-1211z^{2}w^{16}-1211zw^{17}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
36.72.0-18.a.1.12 $36$ $3$ $3$ $0$ $0$

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

Covering curve Level Index Degree Genus
108.432.10-54.a.1.7 $108$ $2$ $2$ $10$
108.432.10-108.a.1.12 $108$ $2$ $2$ $10$
108.432.10-108.b.1.8 $108$ $2$ $2$ $10$
108.432.10-108.c.1.10 $108$ $2$ $2$ $10$
108.432.10-108.j.1.4 $108$ $2$ $2$ $10$
108.432.10-108.k.1.7 $108$ $2$ $2$ $10$
108.432.10-108.l.1.5 $108$ $2$ $2$ $10$
108.432.10-54.n.1.3 $108$ $2$ $2$ $10$
108.432.13-108.q.1.4 $108$ $2$ $2$ $13$
108.432.13-108.r.1.2 $108$ $2$ $2$ $13$
108.432.13-108.s.1.3 $108$ $2$ $2$ $13$
108.432.13-108.t.1.1 $108$ $2$ $2$ $13$
216.432.10-216.a.1.16 $216$ $2$ $2$ $10$
216.432.10-216.b.1.16 $216$ $2$ $2$ $10$
216.432.10-216.c.1.14 $216$ $2$ $2$ $10$
216.432.10-216.d.1.12 $216$ $2$ $2$ $10$
216.432.10-216.q.1.16 $216$ $2$ $2$ $10$
216.432.10-216.r.1.16 $216$ $2$ $2$ $10$
216.432.10-216.s.1.14 $216$ $2$ $2$ $10$
216.432.10-216.t.1.12 $216$ $2$ $2$ $10$
216.432.13-216.bc.1.16 $216$ $2$ $2$ $13$
216.432.13-216.bd.1.16 $216$ $2$ $2$ $13$
216.432.13-216.be.1.16 $216$ $2$ $2$ $13$
216.432.13-216.bf.1.16 $216$ $2$ $2$ $13$