Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $1764$ | ||
| Index: | $256$ | $\PSL_2$-index: | $128$ | ||||
| Genus: | $5 = 1 + \frac{ 128 }{12} - \frac{ 0 }{4} - \frac{ 8 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot6^{2}\cdot14^{2}\cdot42^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $8$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42I5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.256.5.24 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}23&18\\21&25\end{bmatrix}$, $\begin{bmatrix}41&5\\27&28\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.128.5.d.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $3$ |
| Cyclic 42-torsion field degree: | $36$ |
| Full 42-torsion field degree: | $2268$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{7}\cdot7^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 21.2.a.a, 84.2.f.a, 1764.2.a.g, 1764.2.a.k |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 5 x^{2} - 11 x z + x t - z w + w^{2} + 2 w t $ |
| $=$ | $5 x^{2} - x w + 16 z^{2} - z w - z t + 2 w t + t^{2}$ | |
| $=$ | $x z - 2 x w - 2 x t - 21 y^{2} + 2 z w - 4 w^{2} - 4 w t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 103950 x^{8} + 40635 x^{7} y + 5904 x^{6} y^{2} + 1009890 x^{6} z^{2} + 378 x^{5} y^{3} + \cdots + 3495856 z^{8} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(3/11:0:-2/11:-12/11:1)$, $(-3/10:0:1/5:-9/10:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 42.64.3.e.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}Y^{2}-3Y^{4}-2X^{3}Z+8XY^{2}Z-X^{2}Z^{2}-6Y^{2}Z^{2}+6XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 42.128.5.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+2w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 21t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3y$ |
Equation of the image curve:
| $0$ | $=$ | $ 103950X^{8}+40635X^{7}Y+5904X^{6}Y^{2}+378X^{5}Y^{3}+9X^{4}Y^{4}+1009890X^{6}Z^{2}+291942X^{5}YZ^{2}+27909X^{4}Y^{2}Z^{2}+882X^{3}Y^{3}Z^{2}+3659124X^{4}Z^{4}+695604X^{3}YZ^{4}+32830X^{2}Y^{2}Z^{4}+5856039X^{2}Z^{6}+549829XYZ^{6}+3495856Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.128.1-21.a.4.3 | $21$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 42.128.1-21.a.4.3 | $42$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 42.128.3-42.d.1.1 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.128.3-42.d.1.8 | $42$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 42.128.3-42.e.2.2 | $42$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 42.128.3-42.e.2.8 | $42$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.768.21-42.l.3.2 | $42$ | $3$ | $3$ | $21$ | $0$ | $2^{8}$ |
| 42.768.21-42.n.4.1 | $42$ | $3$ | $3$ | $21$ | $0$ | $2^{8}$ |
| 42.768.21-42.p.1.4 | $42$ | $3$ | $3$ | $21$ | $2$ | $1^{8}\cdot4^{2}$ |
| 42.768.21-42.r.1.4 | $42$ | $3$ | $3$ | $21$ | $0$ | $1^{8}\cdot2^{4}$ |
| 42.768.25-42.c.2.2 | $42$ | $3$ | $3$ | $25$ | $3$ | $1^{8}\cdot2^{2}\cdot4^{2}$ |
| 42.1792.57-42.f.2.4 | $42$ | $7$ | $7$ | $57$ | $10$ | $1^{18}\cdot2^{9}\cdot8^{2}$ |