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 | $4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.613 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&8\\10&37\end{bmatrix}$, $\begin{bmatrix}23&36\\28&29\end{bmatrix}$, $\begin{bmatrix}35&36\\4&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.96.1.bs.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| 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.bs.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
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.g.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-8.g.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.i.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.i.1.12 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.j.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.j.1.16 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.z.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.0-40.z.2.10 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 40.96.1-40.bg.1.10 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.bg.1.16 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.bh.2.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.bh.2.16 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.bs.1.7 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.96.1-40.bs.1.13 | $40$ | $2$ | $2$ | $1$ | $0$ | 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.kw.2.8 | $40$ | $5$ | $5$ | $33$ | $9$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.vu.1.13 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.bge.2.6 | $40$ | $10$ | $10$ | $65$ | $11$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |