Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $76$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $2^{3}\cdot38^{3}$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 38A8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.240.8.1 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}7&0\\0&29\end{bmatrix}$, $\begin{bmatrix}23&12\\0&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 38.120.8.a.1 for the level structure with $-I$) |
| Cyclic 38-isogeny field degree: | $1$ |
| Cyclic 38-torsion field degree: | $9$ |
| Full 38-torsion field degree: | $3078$ |
Jacobian
| Conductor: | $2^{6}\cdot19^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{8}$ |
| Newforms: | 19.2.a.a$^{3}$, 38.2.a.a$^{2}$, 38.2.a.b$^{2}$, 76.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x^{2} + x t + x u + z t - t r + u^{2} + u v $ |
| $=$ | $x w - x u - z u - t^{2} + u v + u r$ | |
| $=$ | $x v + y t + z t - t^{2} - t u - t v - t r + u v$ | |
| $=$ | $x y + x w + x u - x r - y^{2} + y v + u^{2} + u r - v^{2} + v r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 64 x^{12} - 352 x^{11} y + 960 x^{10} y^{2} - 80 x^{10} y z + 96 x^{10} z^{2} - 1724 x^{9} y^{3} + \cdots + y^{6} z^{6} $ |
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 |
|---|
| $(1:1:0:1:0:-1:1:1)$, $(1:1:0:0:0:-1:1:0)$, $(0:0:0:0:0:0:0:1)$, $(0:1:2:1:0:0:1:1)$, $(0:1:1:0:0:0:1:1)$, $(0:0:1:0:0:0:0:0)$ |
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 $X_0(38)$ :
| $\displaystyle X$ | $=$ | $\displaystyle x-y+r$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x-t-u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+y-z+r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -x+v-r$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{2}-XY+2XZ-YZ+2XW-YW+ZW $ |
| $=$ | $ Y^{3}-X^{2}Z+XZ^{2}+X^{2}W-XYW+YZW+2XW^{2}+YW^{2}-ZW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 38.120.8.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 64X^{12}-352X^{11}Y+960X^{10}Y^{2}-80X^{10}YZ+96X^{10}Z^{2}-1724X^{9}Y^{3}+296X^{9}Y^{2}Z-352X^{9}YZ^{2}+64X^{9}Z^{3}+2265X^{8}Y^{4}-594X^{8}Y^{3}Z+488X^{8}Y^{2}Z^{2}-280X^{8}YZ^{3}+144X^{8}Z^{4}-2278X^{7}Y^{5}+927X^{7}Y^{4}Z-330X^{7}Y^{3}Z^{2}+280X^{7}Y^{2}Z^{3}-280X^{7}YZ^{4}+48X^{7}Z^{5}+1762X^{6}Y^{6}-1125X^{6}Y^{5}Z+170X^{6}Y^{4}Z^{2}+96X^{6}Y^{3}Z^{3}+68X^{6}Y^{2}Z^{4}-80X^{6}YZ^{5}+16X^{6}Z^{6}-1016X^{5}Y^{7}+949X^{5}Y^{6}Z-198X^{5}Y^{5}Z^{2}-273X^{5}Y^{4}Z^{3}+216X^{5}Y^{3}Z^{4}-4X^{5}Y^{2}Z^{5}-16X^{5}YZ^{6}+412X^{4}Y^{8}-514X^{4}Y^{7}Z+225X^{4}Y^{6}Z^{2}+69X^{4}Y^{5}Z^{3}-175X^{4}Y^{4}Z^{4}+80X^{4}Y^{3}Z^{5}-12X^{4}Y^{2}Z^{6}-108X^{3}Y^{9}+167X^{3}Y^{8}Z-138X^{3}Y^{7}Z^{2}+90X^{3}Y^{6}Z^{3}-40X^{3}Y^{4}Z^{5}+16X^{3}Y^{3}Z^{6}+16X^{2}Y^{10}-29X^{2}Y^{9}Z+42X^{2}Y^{8}Z^{2}-65X^{2}Y^{7}Z^{3}+51X^{2}Y^{6}Z^{4}-14X^{2}Y^{5}Z^{5}-XY^{11}+2XY^{10}Z-5XY^{9}Z^{2}+15XY^{8}Z^{3}-22XY^{7}Z^{4}+15XY^{6}Z^{5}-4XY^{5}Z^{6}-Y^{9}Z^{3}+3Y^{8}Z^{4}-3Y^{7}Z^{5}+Y^{6}Z^{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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $40$ | $20$ | $0$ | $0$ | full Jacobian |
| 19.40.1-19.a.1.2 | $19$ | $6$ | $6$ | $1$ | $0$ | $1^{7}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.80.2-38.a.1.1 | $38$ | $3$ | $3$ | $2$ | $0$ | $1^{6}$ |
| 38.120.4-38.a.1.2 | $38$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
| 38.120.4-38.a.1.3 | $38$ | $2$ | $2$ | $4$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.720.22-38.a.1.1 | $38$ | $3$ | $3$ | $22$ | $0$ | $2^{3}\cdot4^{2}$ |
| 38.720.22-38.a.2.2 | $38$ | $3$ | $3$ | $22$ | $0$ | $2^{3}\cdot4^{2}$ |
| 38.720.22-38.b.1.1 | $38$ | $3$ | $3$ | $22$ | $4$ | $1^{6}\cdot2^{4}$ |
| 38.4560.161-38.a.1.1 | $38$ | $19$ | $19$ | $161$ | $56$ | $1^{29}\cdot2^{23}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$ |
| 76.480.16-76.a.1.2 | $76$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 76.480.16-76.a.2.4 | $76$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 76.480.17-76.a.1.2 | $76$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 76.480.17-76.b.1.2 | $76$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 76.480.17-76.c.1.2 | $76$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 76.480.17-76.d.1.2 | $76$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 76.480.18-76.a.1.6 | $76$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 76.480.18-76.a.2.8 | $76$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 152.480.16-152.a.1.4 | $152$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 152.480.16-152.a.2.8 | $152$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 152.480.17-152.a.1.6 | $152$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 152.480.17-152.b.1.6 | $152$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 152.480.17-152.c.1.8 | $152$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 152.480.17-152.d.1.8 | $152$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 152.480.18-152.a.1.11 | $152$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 152.480.18-152.a.2.14 | $152$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 228.480.16-228.a.1.4 | $228$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 228.480.16-228.a.2.8 | $228$ | $2$ | $2$ | $16$ | $?$ | not computed |
| 228.480.17-228.a.1.3 | $228$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 228.480.17-228.b.1.4 | $228$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 228.480.17-228.c.1.4 | $228$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 228.480.17-228.d.1.4 | $228$ | $2$ | $2$ | $17$ | $?$ | not computed |
| 228.480.18-228.a.1.9 | $228$ | $2$ | $2$ | $18$ | $?$ | not computed |
| 228.480.18-228.a.2.10 | $228$ | $2$ | $2$ | $18$ | $?$ | not computed |