Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $1600$ | ||
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 | $2^{2}\cdot4^{3}$ | ||
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: | 40.192.1.172 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}17&28\\34&37\end{bmatrix}$, $\begin{bmatrix}27&32\\4&37\end{bmatrix}$, $\begin{bmatrix}31&28\\12&37\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.96.1.bu.1 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: | $2^{6}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 1600.2.a.n |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 2 x^{2} - y^{2} - z^{2} $ |
$=$ | $x^{2} + y^{2} + y w + 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} + 3 x^{2} z^{2} + 9 y^{4} + 4 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^8}\cdot\frac{15819935688yz^{22}w-2053015632yz^{20}w^{3}-185379217920yz^{18}w^{5}-180688050432yz^{16}w^{7}+403618723200yz^{14}w^{9}+1083756360960yz^{12}w^{11}+1167579463680yz^{10}w^{13}+733976764416yz^{8}w^{15}+290019502080yz^{6}w^{17}+71232860160yz^{4}w^{19}+9949986816yz^{2}w^{21}+603029504yw^{23}-40209003207z^{24}+217027041228z^{22}w^{2}+5029478892z^{20}w^{4}-855747555840z^{18}w^{6}-734670966816z^{16}w^{8}+929914032960z^{14}w^{10}+2361533952960z^{12}w^{12}+2270183915520z^{10}w^{14}+1283814074880z^{8}w^{16}+461939235840z^{6}w^{18}+104487312384z^{4}w^{20}+13569744896z^{2}w^{22}+770625536w^{24}}{z^{8}(43740yz^{14}w+376164yz^{12}w^{3}+1268460yz^{10}w^{5}+2124792yz^{8}w^{7}+1877040yz^{6}w^{9}+887328yz^{4}w^{11}+212352yz^{2}w^{13}+20224yw^{15}+54675z^{16}+575910z^{14}w^{2}+2499741z^{12}w^{4}+5717790z^{10}w^{6}+7345971z^{8}w^{8}+5315112z^{6}w^{10}+2143128z^{4}w^{12}+449728z^{2}w^{14}+38320w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.bu.1 :
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}+6X^{2}Y^{2}+9Y^{4}+3X^{2}Z^{2}+4Y^{2}Z^{2}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.96.0-8.h.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-8.h.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.i.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.i.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.k.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.k.2.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.y.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.y.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.1-40.bg.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.bg.2.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.bi.2.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.bi.2.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.bs.1.12 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.bs.1.15 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.ky.2.4 | $40$ | $5$ | $5$ | $33$ | $7$ | $1^{14}\cdot2^{9}$ |
40.1152.33-40.vw.2.4 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
40.1920.65-40.bgg.1.15 | $40$ | $10$ | $10$ | $65$ | $9$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |