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Properties

Label 40.192.1-40.bo.2.8
Level $40$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $64$
Index: $192$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $4^{8}\cdot8^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.192.1.1128

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&26\\8&9\end{bmatrix}$, $\begin{bmatrix}15&22\\36&31\end{bmatrix}$, $\begin{bmatrix}31&20\\28&33\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.96.1.bo.2 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $3840$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 2 y^{2} - y z + 2 y w + 2 z w - 2 w^{2} $
$=$ $10 x^{2} - 8 y^{2} - y z + 2 y w - z^{2} + 2 z w - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 7 x^{4} - 16 x^{3} z - 5 x^{2} y^{2} + 18 x^{2} z^{2} + 20 x y^{2} z - 4 x z^{3} - 20 y^{2} z^{2} + 2 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^2}{5^4}\cdot\frac{137451211065yz^{23}-8962356111030yz^{22}w+272201290076700yz^{21}w^{2}-5174768516257800yz^{20}w^{3}+69528031589250000yz^{19}w^{4}-704580644093580000yz^{18}w^{5}+5599877659409880000yz^{17}w^{6}-35785384128265680000yz^{16}w^{7}+186858584930520000000yz^{15}w^{8}-805473661170960000000yz^{14}w^{9}+2883635004777120000000yz^{13}w^{10}-8598201910683840000000yz^{12}w^{11}+21358270084694400000000yz^{11}w^{12}-44114217067065600000000yz^{10}w^{13}+75461015725632000000000yz^{9}w^{14}-106253227242777600000000yz^{8}w^{15}+122084772410880000000000yz^{7}w^{16}-113087661511680000000000yz^{6}w^{17}+83021893171200000000000yz^{5}w^{18}-47127399321600000000000yz^{4}w^{19}+19925331148800000000000yz^{3}w^{20}-5901867417600000000000yz^{2}w^{21}+1092059136000000000000yzw^{22}-94961664000000000000yw^{23}+8388608z^{24}-274902422130z^{23}w+16552213377330z^{22}w^{2}-468504007490400z^{21}w^{3}+8359281340735200z^{20}w^{4}-105965248656900000z^{19}w^{5}+1016911160703820000z^{18}w^{6}-7673796762212160000z^{17}w^{7}+46641809796353040000z^{16}w^{8}-231898066400240000000z^{15}w^{9}+952392221170608000000z^{14}w^{10}-3249393688458240000000z^{13}w^{11}+9234049901581440000000z^{12}w^{12}-21860493779884800000000z^{11}w^{13}+43030628489606400000000z^{10}w^{14}-70156336269824000000000z^{9}w^{15}+94174773204288000000000z^{8}w^{16}-103198303272960000000000z^{7}w^{17}+91217002506240000000000z^{6}w^{18}-63941218406400000000000z^{5}w^{19}+34681614551040000000000z^{4}w^{20}-14021521817600000000000z^{3}w^{21}+3974402457600000000000z^{2}w^{22}-704274432000000000000zw^{23}+58689536000000000000w^{24}}{z^{4}(321489yz^{19}-26300862yz^{18}w+950140692yz^{17}w^{2}-20183504184yz^{16}w^{3}+283509823680yz^{15}w^{4}-2808783437184yz^{14}w^{5}+20441002401792yz^{13}w^{6}-112310736755712yz^{12}w^{7}+474759177004800yz^{11}w^{8}-1563765154214400yz^{10}w^{9}+4044491072793600yz^{9}w^{10}-8240632975411200yz^{8}w^{11}+13211126043648000yz^{7}w^{12}-16562574292992000yz^{6}w^{13}+16038699724800000yz^{5}w^{14}-11748127088640000yz^{4}w^{15}+6288657408000000yz^{3}w^{16}-2319280128000000yz^{2}w^{17}+526479360000000yzw^{18}-55418880000000yw^{19}-642978z^{19}w+49386834z^{18}w^{2}-1669421664z^{17}w^{3}+33109900848z^{16}w^{4}-433781464320z^{15}w^{5}+4008164525568z^{14}w^{6}-27223664914944z^{13}w^{7}+139758934718336z^{12}w^{8}-552794060761600z^{11}w^{9}+1706367673489920z^{10}w^{10}-4142639329587200z^{9}w^{11}+7935662896179200z^{8}w^{12}-11979804338176000z^{7}w^{13}+14163629447168000z^{6}w^{14}-12952971673600000z^{5}w^{15}+8972312883200000z^{4}w^{16}-4547536896000000z^{3}w^{17}+1589889024000000z^{2}w^{18}-342507520000000zw^{19}+34250752000000w^{20})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.bo.2 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ 7X^{4}-5X^{2}Y^{2}-16X^{3}Z+20XY^{2}Z+18X^{2}Z^{2}-20Y^{2}Z^{2}-4XZ^{3}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1-8.n.1.6 $8$ $2$ $2$ $1$ $0$ dimension zero
40.96.0-40.o.2.8 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.o.2.9 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.p.1.8 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.p.1.13 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.w.2.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.w.2.12 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.x.1.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.x.1.10 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.1-8.n.1.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.x.2.14 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.x.2.15 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.ba.1.7 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.ba.1.11 $40$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.960.33-40.do.2.4 $40$ $5$ $5$ $33$ $5$ $1^{14}\cdot2^{9}$
40.1152.33-40.mg.1.9 $40$ $6$ $6$ $33$ $3$ $1^{14}\cdot2\cdot4^{4}$
40.1920.65-40.rq.2.1 $40$ $10$ $10$ $65$ $9$ $1^{28}\cdot2^{10}\cdot4^{4}$