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Properties

Label 88.144.4-22.a.1.5
Level $88$
Index $144$
Genus $4$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $88$ $\SL_2$-level: $44$ Newform level: $44$
Index: $144$ $\PSL_2$-index:$72$
Genus: $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{3}\cdot22^{3}$ Cusp orbits $1^{6}$
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: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 22A4

Level structure

$\GL_2(\Z/88\Z)$-generators: $\begin{bmatrix}3&38\\54&41\end{bmatrix}$, $\begin{bmatrix}15&20\\70&65\end{bmatrix}$, $\begin{bmatrix}29&0\\30&15\end{bmatrix}$, $\begin{bmatrix}47&6\\24&65\end{bmatrix}$, $\begin{bmatrix}49&84\\80&1\end{bmatrix}$, $\begin{bmatrix}55&58\\18&15\end{bmatrix}$
Contains $-I$: no $\quad$ (see 22.72.4.a.1 for the level structure with $-I$)
Cyclic 88-isogeny field degree: $4$
Cyclic 88-torsion field degree: $160$
Full 88-torsion field degree: $140800$

Models

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

$ 0 $ $=$ $ x^{2} + 2 x y + x z + 4 y^{2} + 2 y z - 2 z w - w^{2} $
$=$ $x^{2} y + 2 x y^{2} + x y z - 2 y^{2} z + 2 y z w + y w^{2} + 2 z^{2} w + z w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 4 x^{4} z^{2} - 6 x^{3} y^{2} z + 4 x^{3} y z^{2} + 6 x^{3} z^{3} - 2 x^{2} y^{4} + 3 x^{2} y^{3} z + \cdots + 2 z^{6} $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

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 -2^8\,\frac{523440xy^{2}z^{9}+19053360xy^{2}z^{8}w+43801920xy^{2}z^{7}w^{2}+65448000xy^{2}z^{6}w^{3}+48486240xy^{2}z^{5}w^{4}-37452960xy^{2}z^{4}w^{5}-90659520xy^{2}z^{3}w^{6}-60868800xy^{2}z^{2}w^{7}-17969040xy^{2}zw^{8}-1996560xy^{2}w^{9}-3543120xyz^{10}+3282480xyz^{9}w+48985920xyz^{8}w^{2}+118670400xyz^{7}w^{3}+130065120xyz^{6}w^{4}+56325600xyz^{5}w^{5}-11672640xyz^{4}w^{6}-20831040xyz^{3}w^{7}-7186320xyz^{2}w^{8}-798480xyzw^{9}-1771560xz^{11}-3804480xz^{10}w+4692060xz^{9}w^{2}+36380340xz^{8}w^{3}+58972320xz^{7}w^{4}+63118800xz^{6}w^{5}+99169560xz^{5}w^{6}+133485480xz^{4}w^{7}+103205880xz^{3}w^{8}+43945920xz^{2}w^{9}+9743580xzw^{10}+885780xw^{11}+391816y^{2}z^{10}-2881384y^{2}z^{9}w+21476960y^{2}z^{8}w^{2}+68973920y^{2}z^{7}w^{3}+39077344y^{2}z^{6}w^{4}+14332368y^{2}z^{5}w^{5}+75195152y^{2}z^{4}w^{6}+102749472y^{2}z^{3}w^{7}+57820824y^{2}z^{2}w^{8}+14992920y^{2}zw^{9}+1479312y^{2}w^{10}-3543120yz^{11}+6891064yz^{10}w+82928860yz^{9}w^{2}+188233388yz^{8}w^{3}+190158736yz^{7}w^{4}+100678240yz^{6}w^{5}+32755896yz^{5}w^{6}+7463032yz^{4}w^{7}-1769776yz^{3}w^{8}-2992200yz^{2}w^{9}-1051044yzw^{10}-112500yw^{11}-1771561z^{12}-3739028z^{11}w+20928944z^{10}w^{2}+65671018z^{9}w^{3}+67202953z^{8}w^{4}+22013432z^{7}w^{5}-71561420z^{6}w^{6}-192512524z^{5}w^{7}-226696019z^{4}w^{8}-146720620z^{3}w^{9}-54042300z^{2}w^{10}-10685622zw^{11}-885781w^{12}}{174080xy^{2}z^{9}+630016xy^{2}z^{8}w+1233152xy^{2}z^{7}w^{2}+1617792xy^{2}z^{6}w^{3}+1530080xy^{2}z^{5}w^{4}+1138400xy^{2}z^{4}w^{5}+658112xy^{2}z^{3}w^{6}+266816xy^{2}z^{2}w^{7}+64800xy^{2}zw^{8}+7200xy^{2}w^{9}-87424xyz^{9}w-353344xyz^{8}w^{2}-761536xyz^{7}w^{3}-1070848xyz^{6}w^{4}-999424xyz^{5}w^{5}-622720xyz^{4}w^{6}-253312xyz^{3}w^{7}-61632xyz^{2}w^{8}-6848xyzw^{9}-87168xz^{10}w-313984xz^{9}w^{2}-593280xz^{8}w^{3}-770192xz^{7}w^{4}-707296xz^{6}w^{5}-451080xz^{5}w^{6}-189048xz^{4}w^{7}-46728xz^{3}w^{8}-5192xz^{2}w^{9}+164096y^{2}z^{10}+1114368y^{2}z^{9}w+3198976y^{2}z^{8}w^{2}+5755584y^{2}z^{7}w^{3}+7004192y^{2}z^{6}w^{4}+5895632y^{2}z^{5}w^{5}+3457088y^{2}z^{4}w^{6}+1418528y^{2}z^{3}w^{7}+412184y^{2}z^{2}w^{8}+83072y^{2}zw^{9}+9352y^{2}w^{10}-82304yz^{10}w-596800yz^{9}w^{2}-1848864yz^{8}w^{3}-3559312yz^{7}w^{4}-4586608yz^{6}w^{5}-3931456yz^{5}w^{6}-2148872yz^{4}w^{7}-673632yz^{3}w^{8}-80208yz^{2}w^{9}+12620yzw^{10}+3600yw^{11}-82048z^{11}w-557568z^{10}w^{2}-1574304z^{9}w^{3}-2763712z^{8}w^{4}-3283424z^{7}w^{5}-2639860z^{6}w^{6}-1388860z^{5}w^{7}-437807z^{4}w^{8}-61452z^{3}w^{9}+4004z^{2}w^{10}+1800zw^{11}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 22.72.4.a.1 :

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

Equation of the image curve:

$0$ $=$ $ -4X^{4}Z^{2}-6X^{3}Y^{2}Z+4X^{3}YZ^{2}+6X^{3}Z^{3}-2X^{2}Y^{4}+3X^{2}Y^{3}Z+8X^{2}Y^{2}Z^{2}-16X^{2}YZ^{3}+2X^{2}Z^{4}+XY^{5}-9XY^{3}Z^{2}-4XY^{2}Z^{3}+20XYZ^{4}-6XZ^{5}-Y^{6}-2Y^{5}Z+6Y^{3}Z^{3}+2Y^{2}Z^{4}-8YZ^{5}+2Z^{6} $

Modular covers

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

Factor curve Level Index Degree Genus Rank
8.12.0-2.a.1.1 $8$ $12$ $12$ $0$ $0$
$X_0(11)$ $11$ $12$ $6$ $1$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.12.0-2.a.1.1 $8$ $12$ $12$ $0$ $0$

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

Covering curve Level Index Degree Genus
88.288.8-44.a.1.1 $88$ $2$ $2$ $8$
88.288.8-44.a.1.4 $88$ $2$ $2$ $8$
88.288.8-44.a.1.5 $88$ $2$ $2$ $8$
88.288.8-44.a.1.8 $88$ $2$ $2$ $8$
88.288.8-44.a.1.10 $88$ $2$ $2$ $8$
88.288.8-44.a.1.11 $88$ $2$ $2$ $8$
88.288.8-44.a.2.1 $88$ $2$ $2$ $8$
88.288.8-44.a.2.4 $88$ $2$ $2$ $8$
88.288.8-44.a.2.5 $88$ $2$ $2$ $8$
88.288.8-44.a.2.8 $88$ $2$ $2$ $8$
88.288.8-44.a.2.10 $88$ $2$ $2$ $8$
88.288.8-44.a.2.11 $88$ $2$ $2$ $8$
88.288.8-88.a.1.7 $88$ $2$ $2$ $8$
88.288.8-88.a.1.8 $88$ $2$ $2$ $8$
88.288.8-88.a.1.11 $88$ $2$ $2$ $8$
88.288.8-88.a.1.12 $88$ $2$ $2$ $8$
88.288.8-88.a.1.17 $88$ $2$ $2$ $8$
88.288.8-88.a.1.18 $88$ $2$ $2$ $8$
88.288.8-88.a.2.6 $88$ $2$ $2$ $8$
88.288.8-88.a.2.8 $88$ $2$ $2$ $8$
88.288.8-88.a.2.13 $88$ $2$ $2$ $8$
88.288.8-88.a.2.15 $88$ $2$ $2$ $8$
88.288.8-88.a.2.17 $88$ $2$ $2$ $8$
88.288.8-88.a.2.19 $88$ $2$ $2$ $8$
88.288.9-44.a.1.1 $88$ $2$ $2$ $9$
88.288.9-44.a.1.15 $88$ $2$ $2$ $9$
88.288.9-88.a.1.1 $88$ $2$ $2$ $9$
88.288.9-88.a.1.3 $88$ $2$ $2$ $9$
88.288.9-88.a.1.20 $88$ $2$ $2$ $9$
88.288.9-44.b.1.6 $88$ $2$ $2$ $9$
88.288.9-44.b.1.24 $88$ $2$ $2$ $9$
88.288.9-44.b.1.36 $88$ $2$ $2$ $9$
88.288.9-88.b.1.1 $88$ $2$ $2$ $9$
88.288.9-88.b.1.3 $88$ $2$ $2$ $9$
88.288.9-88.b.1.20 $88$ $2$ $2$ $9$
88.288.9-44.c.1.1 $88$ $2$ $2$ $9$
88.288.9-44.c.1.4 $88$ $2$ $2$ $9$
88.288.9-44.c.1.7 $88$ $2$ $2$ $9$
88.288.9-88.c.1.1 $88$ $2$ $2$ $9$
88.288.9-88.c.1.3 $88$ $2$ $2$ $9$
88.288.9-88.c.1.18 $88$ $2$ $2$ $9$
88.288.9-44.d.1.1 $88$ $2$ $2$ $9$
88.288.9-44.d.1.4 $88$ $2$ $2$ $9$
88.288.9-44.d.1.9 $88$ $2$ $2$ $9$
88.288.9-88.d.1.1 $88$ $2$ $2$ $9$
88.288.9-88.d.1.3 $88$ $2$ $2$ $9$
88.288.9-88.d.1.18 $88$ $2$ $2$ $9$
88.288.10-44.a.1.1 $88$ $2$ $2$ $10$
88.288.10-44.a.1.4 $88$ $2$ $2$ $10$
88.288.10-44.a.2.1 $88$ $2$ $2$ $10$
88.288.10-44.a.2.4 $88$ $2$ $2$ $10$
88.288.10-88.a.1.3 $88$ $2$ $2$ $10$
88.288.10-88.a.1.7 $88$ $2$ $2$ $10$
88.288.10-88.a.2.3 $88$ $2$ $2$ $10$
88.288.10-88.a.2.7 $88$ $2$ $2$ $10$
264.288.8-132.a.1.1 $264$ $2$ $2$ $8$
264.288.8-132.a.1.6 $264$ $2$ $2$ $8$
264.288.8-132.a.1.11 $264$ $2$ $2$ $8$
264.288.8-132.a.1.16 $264$ $2$ $2$ $8$
264.288.8-132.a.1.17 $264$ $2$ $2$ $8$
264.288.8-132.a.1.22 $264$ $2$ $2$ $8$
264.288.8-132.a.2.1 $264$ $2$ $2$ $8$
264.288.8-132.a.2.4 $264$ $2$ $2$ $8$
264.288.8-132.a.2.13 $264$ $2$ $2$ $8$
264.288.8-132.a.2.16 $264$ $2$ $2$ $8$
264.288.8-132.a.2.17 $264$ $2$ $2$ $8$
264.288.8-132.a.2.20 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.14 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.15 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.22 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.23 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.33 $264$ $2$ $2$ $8$
264.288.8-264.ea.1.36 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.12 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.14 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.27 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.29 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.33 $264$ $2$ $2$ $8$
264.288.8-264.ea.2.39 $264$ $2$ $2$ $8$
264.288.9-132.a.1.1 $264$ $2$ $2$ $9$
264.288.9-132.a.1.11 $264$ $2$ $2$ $9$
264.288.9-132.a.1.18 $264$ $2$ $2$ $9$
264.288.9-132.b.1.1 $264$ $2$ $2$ $9$
264.288.9-132.b.1.3 $264$ $2$ $2$ $9$
264.288.9-132.b.1.13 $264$ $2$ $2$ $9$
264.288.9-132.c.1.1 $264$ $2$ $2$ $9$
264.288.9-132.c.1.8 $264$ $2$ $2$ $9$
264.288.9-132.c.1.13 $264$ $2$ $2$ $9$
264.288.9-132.d.1.1 $264$ $2$ $2$ $9$
264.288.9-132.d.1.9 $264$ $2$ $2$ $9$
264.288.9-132.d.1.18 $264$ $2$ $2$ $9$
264.288.9-264.bbs.1.3 $264$ $2$ $2$ $9$
264.288.9-264.bbs.1.5 $264$ $2$ $2$ $9$
264.288.9-264.bbs.1.40 $264$ $2$ $2$ $9$
264.288.9-264.bbt.1.5 $264$ $2$ $2$ $9$
264.288.9-264.bbt.1.7 $264$ $2$ $2$ $9$
264.288.9-264.bbt.1.36 $264$ $2$ $2$ $9$
264.288.9-264.bbu.1.2 $264$ $2$ $2$ $9$
264.288.9-264.bbu.1.5 $264$ $2$ $2$ $9$
264.288.9-264.bbu.1.36 $264$ $2$ $2$ $9$
264.288.9-264.bbv.1.3 $264$ $2$ $2$ $9$
264.288.9-264.bbv.1.5 $264$ $2$ $2$ $9$
264.288.9-264.bbv.1.34 $264$ $2$ $2$ $9$
264.288.10-132.a.1.4 $264$ $2$ $2$ $10$
264.288.10-132.a.1.11 $264$ $2$ $2$ $10$
264.288.10-132.a.2.8 $264$ $2$ $2$ $10$
264.288.10-132.a.2.11 $264$ $2$ $2$ $10$
264.288.10-264.e.1.4 $264$ $2$ $2$ $10$
264.288.10-264.e.1.7 $264$ $2$ $2$ $10$
264.288.10-264.e.2.7 $264$ $2$ $2$ $10$
264.288.10-264.e.2.13 $264$ $2$ $2$ $10$
264.432.16-66.a.1.15 $264$ $3$ $3$ $16$