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Properties

Label 80.120.4-40.bl.1.5
Level $80$
Index $120$
Genus $4$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $80$ $\SL_2$-level: $80$ Newform level: $200$
Index: $120$ $\PSL_2$-index:$60$
Genus: $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $5^{2}\cdot10\cdot40$ Cusp orbits $1^{4}$
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: 40B4

Level structure

$\GL_2(\Z/80\Z)$-generators: $\begin{bmatrix}0&69\\71&14\end{bmatrix}$, $\begin{bmatrix}25&72\\44&5\end{bmatrix}$, $\begin{bmatrix}38&39\\77&0\end{bmatrix}$, $\begin{bmatrix}42&43\\19&34\end{bmatrix}$, $\begin{bmatrix}62&37\\1&58\end{bmatrix}$, $\begin{bmatrix}76&51\\45&74\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.60.4.bl.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree: $12$
Cyclic 80-torsion field degree: $192$
Full 80-torsion field degree: $98304$

Models

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

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

$ 0 $ $=$ $ 4 x^{6} - 2 x^{4} y^{2} + 9 x^{4} y z + 5 x^{4} z^{2} + 2 x^{2} y^{4} + 3 x^{2} y^{2} z^{2} + \cdots + y^{3} z^{3} $
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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
$(0:1:0:0)$, $(0:0:0:1)$, $(1:-2:1:1)$, $(-1:-2:-1:1)$

Maps to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{247959xyz^{7}w-1069305xyz^{5}w^{3}+285486xyz^{3}w^{5}+106620xyzw^{7}-362824xz^{9}-2275690xz^{7}w^{2}+1375404xz^{5}w^{4}-604018xz^{3}w^{6}+223160xzw^{8}+128y^{10}+1280y^{9}w+5760y^{8}w^{2}+12800y^{7}w^{3}+8960y^{6}w^{4}-23040y^{5}w^{5}-75520y^{4}w^{6}-102400y^{3}w^{7}-67200y^{2}w^{8}-1499362yz^{8}w+1140415yz^{6}w^{3}+19449yz^{4}w^{5}-23530yz^{2}w^{7}-17540yw^{9}-377232z^{10}+1996906z^{8}w^{2}-2345416z^{6}w^{4}+633222z^{4}w^{6}+152680z^{2}w^{8}+32w^{10}}{8xyz^{7}w-15xyz^{5}w^{3}+4xyz^{3}w^{5}+xyzw^{7}+32xz^{9}-40xz^{7}w^{2}+18xz^{5}w^{4}+2xz^{3}w^{6}-6xzw^{8}+16yz^{8}w+10yz^{6}w^{3}-25yz^{4}w^{5}+10yz^{2}w^{7}+yw^{9}+32z^{8}w^{2}-42z^{6}w^{4}+26z^{4}w^{6}-2z^{2}w^{8}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.60.4.bl.1 :

$\displaystyle X$ $=$ $\displaystyle x+z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Equation of the image curve:

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

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
$X_{S_4}(5)$ $5$ $24$ $12$ $0$ $0$
16.24.0-8.n.1.8 $16$ $5$ $5$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
16.24.0-8.n.1.8 $16$ $5$ $5$ $0$ $0$

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

Covering curve Level Index Degree Genus
80.240.8-40.v.1.7 $80$ $2$ $2$ $8$
80.240.8-40.z.1.7 $80$ $2$ $2$ $8$
80.240.8-40.bj.1.8 $80$ $2$ $2$ $8$
80.240.8-40.bk.1.7 $80$ $2$ $2$ $8$
80.240.8-40.cj.1.2 $80$ $2$ $2$ $8$
80.240.8-40.cl.1.8 $80$ $2$ $2$ $8$
80.240.8-40.cn.1.3 $80$ $2$ $2$ $8$
80.240.8-40.cp.1.8 $80$ $2$ $2$ $8$
80.240.8-40.da.1.1 $80$ $2$ $2$ $8$
80.240.8-40.da.1.9 $80$ $2$ $2$ $8$
80.240.8-40.da.2.1 $80$ $2$ $2$ $8$
80.240.8-40.da.2.3 $80$ $2$ $2$ $8$
80.240.8-40.db.1.1 $80$ $2$ $2$ $8$
80.240.8-40.db.1.9 $80$ $2$ $2$ $8$
80.240.8-40.db.2.1 $80$ $2$ $2$ $8$
80.240.8-40.db.2.3 $80$ $2$ $2$ $8$
80.240.8-40.dc.1.1 $80$ $2$ $2$ $8$
80.240.8-40.dc.1.9 $80$ $2$ $2$ $8$
80.240.8-40.dc.2.1 $80$ $2$ $2$ $8$
80.240.8-40.dc.2.3 $80$ $2$ $2$ $8$
80.240.8-40.dd.1.1 $80$ $2$ $2$ $8$
80.240.8-40.dd.1.9 $80$ $2$ $2$ $8$
80.240.8-40.dd.2.1 $80$ $2$ $2$ $8$
80.240.8-40.dd.2.3 $80$ $2$ $2$ $8$
80.360.10-40.cj.1.2 $80$ $3$ $3$ $10$
80.480.13-40.of.1.2 $80$ $4$ $4$ $13$
80.240.8-80.q.1.1 $80$ $2$ $2$ $8$
80.240.8-80.q.1.17 $80$ $2$ $2$ $8$
80.240.8-80.q.2.1 $80$ $2$ $2$ $8$
80.240.8-80.q.2.9 $80$ $2$ $2$ $8$
80.240.8-80.r.1.1 $80$ $2$ $2$ $8$
80.240.8-80.r.1.17 $80$ $2$ $2$ $8$
80.240.8-80.r.2.1 $80$ $2$ $2$ $8$
80.240.8-80.r.2.9 $80$ $2$ $2$ $8$
80.240.8-80.s.1.1 $80$ $2$ $2$ $8$
80.240.8-80.s.1.17 $80$ $2$ $2$ $8$
80.240.8-80.s.2.1 $80$ $2$ $2$ $8$
80.240.8-80.s.2.9 $80$ $2$ $2$ $8$
80.240.8-80.t.1.1 $80$ $2$ $2$ $8$
80.240.8-80.t.1.17 $80$ $2$ $2$ $8$
80.240.8-80.t.2.1 $80$ $2$ $2$ $8$
80.240.8-80.t.2.9 $80$ $2$ $2$ $8$
80.240.8-80.u.1.3 $80$ $2$ $2$ $8$
80.240.8-80.u.1.19 $80$ $2$ $2$ $8$
80.240.8-80.v.1.5 $80$ $2$ $2$ $8$
80.240.8-80.v.1.21 $80$ $2$ $2$ $8$
80.240.8-80.y.1.3 $80$ $2$ $2$ $8$
80.240.8-80.y.1.19 $80$ $2$ $2$ $8$
80.240.8-80.z.1.5 $80$ $2$ $2$ $8$
80.240.8-80.z.1.21 $80$ $2$ $2$ $8$
80.240.9-80.a.1.5 $80$ $2$ $2$ $9$
80.240.9-80.a.1.21 $80$ $2$ $2$ $9$
80.240.9-80.b.1.3 $80$ $2$ $2$ $9$
80.240.9-80.b.1.19 $80$ $2$ $2$ $9$
80.240.9-80.e.1.5 $80$ $2$ $2$ $9$
80.240.9-80.e.1.21 $80$ $2$ $2$ $9$
80.240.9-80.f.1.3 $80$ $2$ $2$ $9$
80.240.9-80.f.1.19 $80$ $2$ $2$ $9$
240.240.8-120.db.1.14 $240$ $2$ $2$ $8$
240.240.8-120.dd.1.15 $240$ $2$ $2$ $8$
240.240.8-120.df.1.16 $240$ $2$ $2$ $8$
240.240.8-120.dh.1.15 $240$ $2$ $2$ $8$
240.240.8-120.fb.1.5 $240$ $2$ $2$ $8$
240.240.8-120.fd.1.7 $240$ $2$ $2$ $8$
240.240.8-120.ff.1.5 $240$ $2$ $2$ $8$
240.240.8-120.fh.1.7 $240$ $2$ $2$ $8$
240.240.8-120.gg.1.11 $240$ $2$ $2$ $8$
240.240.8-120.gg.1.27 $240$ $2$ $2$ $8$
240.240.8-120.gg.2.11 $240$ $2$ $2$ $8$
240.240.8-120.gg.2.27 $240$ $2$ $2$ $8$
240.240.8-120.gh.1.3 $240$ $2$ $2$ $8$
240.240.8-120.gh.1.19 $240$ $2$ $2$ $8$
240.240.8-120.gh.2.3 $240$ $2$ $2$ $8$
240.240.8-120.gh.2.19 $240$ $2$ $2$ $8$
240.240.8-120.gi.1.3 $240$ $2$ $2$ $8$
240.240.8-120.gi.1.19 $240$ $2$ $2$ $8$
240.240.8-120.gi.2.3 $240$ $2$ $2$ $8$
240.240.8-120.gi.2.19 $240$ $2$ $2$ $8$
240.240.8-120.gj.1.11 $240$ $2$ $2$ $8$
240.240.8-120.gj.1.27 $240$ $2$ $2$ $8$
240.240.8-120.gj.2.11 $240$ $2$ $2$ $8$
240.240.8-120.gj.2.27 $240$ $2$ $2$ $8$
240.360.14-120.fd.1.63 $240$ $3$ $3$ $14$
240.480.17-120.brb.1.28 $240$ $4$ $4$ $17$
240.240.8-240.q.1.4 $240$ $2$ $2$ $8$
240.240.8-240.q.1.8 $240$ $2$ $2$ $8$
240.240.8-240.q.2.4 $240$ $2$ $2$ $8$
240.240.8-240.q.2.8 $240$ $2$ $2$ $8$
240.240.8-240.r.1.4 $240$ $2$ $2$ $8$
240.240.8-240.r.1.8 $240$ $2$ $2$ $8$
240.240.8-240.r.2.4 $240$ $2$ $2$ $8$
240.240.8-240.r.2.8 $240$ $2$ $2$ $8$
240.240.8-240.s.1.4 $240$ $2$ $2$ $8$
240.240.8-240.s.1.8 $240$ $2$ $2$ $8$
240.240.8-240.s.2.4 $240$ $2$ $2$ $8$
240.240.8-240.s.2.12 $240$ $2$ $2$ $8$
240.240.8-240.t.1.4 $240$ $2$ $2$ $8$
240.240.8-240.t.1.8 $240$ $2$ $2$ $8$
240.240.8-240.t.2.4 $240$ $2$ $2$ $8$
240.240.8-240.t.2.12 $240$ $2$ $2$ $8$
240.240.8-240.u.1.2 $240$ $2$ $2$ $8$
240.240.8-240.u.1.6 $240$ $2$ $2$ $8$
240.240.8-240.v.1.2 $240$ $2$ $2$ $8$
240.240.8-240.v.1.6 $240$ $2$ $2$ $8$
240.240.8-240.y.1.2 $240$ $2$ $2$ $8$
240.240.8-240.y.1.4 $240$ $2$ $2$ $8$
240.240.8-240.z.1.2 $240$ $2$ $2$ $8$
240.240.8-240.z.1.4 $240$ $2$ $2$ $8$
240.240.9-240.a.1.2 $240$ $2$ $2$ $9$
240.240.9-240.a.1.4 $240$ $2$ $2$ $9$
240.240.9-240.b.1.2 $240$ $2$ $2$ $9$
240.240.9-240.b.1.6 $240$ $2$ $2$ $9$
240.240.9-240.e.1.4 $240$ $2$ $2$ $9$
240.240.9-240.e.1.8 $240$ $2$ $2$ $9$
240.240.9-240.f.1.4 $240$ $2$ $2$ $9$
240.240.9-240.f.1.12 $240$ $2$ $2$ $9$