Invariants
| Level: | $4$ | $\SL_2$-level: | $4$ | ||||
| Index: | $4$ | $\PSL_2$-index: | $4$ | ||||
| Genus: | $0 = 1 + \frac{ 4 }{12} - \frac{ 2 }{4} - \frac{ 1 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $4$ | Cusp orbits | $1$ | ||
| Elliptic points: | $2$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-4,-11,-19,-27,-43,-67,-163$) |
Other labels
| Cummins and Pauli (CP) label: | 4A0 |
| Sutherland and Zywina (SZ) label: | 4A0-4a |
| Rouse and Zureick-Brown (RZB) label: | X7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 4.4.0.1 |
Level structure
| $\GL_2(\Z/4\Z)$-generators: | $\begin{bmatrix}0&1\\1&0\end{bmatrix}$, $\begin{bmatrix}1&3\\2&3\end{bmatrix}$ |
| $\GL_2(\Z/4\Z)$-subgroup: | $C_3:D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 4-isogeny field degree: | $6$ |
| Cyclic 4-torsion field degree: | $12$ |
| Full 4-torsion field degree: | $24$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 454 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 4 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{x^{4}(x-12y)(x+4y)^{3}}{y^{4}x^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| $X_{\mathrm{ns}}(4)$ | $4$ | $2$ | $2$ | $0$ |
| 4.8.0.b.1 | $4$ | $2$ | $2$ | $0$ |
| 4.12.0.e.1 | $4$ | $3$ | $3$ | $0$ |
| 8.8.0.a.1 | $8$ | $2$ | $2$ | $0$ |
| 8.8.0.b.1 | $8$ | $2$ | $2$ | $0$ |
| $X_{\mathrm{ns}}^+(8)$ | $8$ | $4$ | $4$ | $0$ |
| 12.8.0.c.1 | $12$ | $2$ | $2$ | $0$ |
| 12.8.0.d.1 | $12$ | $2$ | $2$ | $0$ |
| 12.12.0.q.1 | $12$ | $3$ | $3$ | $0$ |
| 12.16.1.a.1 | $12$ | $4$ | $4$ | $1$ |
| 20.8.0.a.1 | $20$ | $2$ | $2$ | $0$ |
| 20.8.0.b.1 | $20$ | $2$ | $2$ | $0$ |
| 20.20.1.c.1 | $20$ | $5$ | $5$ | $1$ |
| 20.24.1.g.1 | $20$ | $6$ | $6$ | $1$ |
| 20.40.2.e.1 | $20$ | $10$ | $10$ | $2$ |
| 24.8.0.e.1 | $24$ | $2$ | $2$ | $0$ |
| 24.8.0.f.1 | $24$ | $2$ | $2$ | $0$ |
| 28.8.0.a.1 | $28$ | $2$ | $2$ | $0$ |
| 28.8.0.b.1 | $28$ | $2$ | $2$ | $0$ |
| 28.32.2.a.1 | $28$ | $8$ | $8$ | $2$ |
| 28.84.4.a.1 | $28$ | $21$ | $21$ | $4$ |
| 28.112.6.a.1 | $28$ | $28$ | $28$ | $6$ |
| 36.108.6.g.1 | $36$ | $27$ | $27$ | $6$ |
| 40.8.0.a.1 | $40$ | $2$ | $2$ | $0$ |
| 40.8.0.b.1 | $40$ | $2$ | $2$ | $0$ |
| 44.8.0.a.1 | $44$ | $2$ | $2$ | $0$ |
| 44.8.0.b.1 | $44$ | $2$ | $2$ | $0$ |
| 44.48.4.c.1 | $44$ | $12$ | $12$ | $4$ |
| 44.220.13.m.1 | $44$ | $55$ | $55$ | $13$ |
| 44.220.14.a.1 | $44$ | $55$ | $55$ | $14$ |
| 44.264.17.m.1 | $44$ | $66$ | $66$ | $17$ |
| 52.8.0.a.1 | $52$ | $2$ | $2$ | $0$ |
| 52.8.0.b.1 | $52$ | $2$ | $2$ | $0$ |
| 52.56.3.a.1 | $52$ | $14$ | $14$ | $3$ |
| 52.312.21.m.1 | $52$ | $78$ | $78$ | $21$ |
| 52.364.24.a.1 | $52$ | $91$ | $91$ | $24$ |
| 52.364.25.a.1 | $52$ | $91$ | $91$ | $25$ |
| 56.8.0.a.1 | $56$ | $2$ | $2$ | $0$ |
| 56.8.0.b.1 | $56$ | $2$ | $2$ | $0$ |
| 60.8.0.c.1 | $60$ | $2$ | $2$ | $0$ |
| 60.8.0.d.1 | $60$ | $2$ | $2$ | $0$ |
| 68.8.0.a.1 | $68$ | $2$ | $2$ | $0$ |
| 68.8.0.b.1 | $68$ | $2$ | $2$ | $0$ |
| 68.72.5.i.1 | $68$ | $18$ | $18$ | $5$ |
| 68.544.38.a.1 | $68$ | $136$ | $136$ | $38$ |
| 68.612.43.o.1 | $68$ | $153$ | $153$ | $43$ |
| 76.8.0.a.1 | $76$ | $2$ | $2$ | $0$ |
| 76.8.0.b.1 | $76$ | $2$ | $2$ | $0$ |
| 76.80.6.a.1 | $76$ | $20$ | $20$ | $6$ |
| 84.8.0.c.1 | $84$ | $2$ | $2$ | $0$ |
| 84.8.0.d.1 | $84$ | $2$ | $2$ | $0$ |
| 88.8.0.a.1 | $88$ | $2$ | $2$ | $0$ |
| 88.8.0.b.1 | $88$ | $2$ | $2$ | $0$ |
| 92.8.0.a.1 | $92$ | $2$ | $2$ | $0$ |
| 92.8.0.b.1 | $92$ | $2$ | $2$ | $0$ |
| 92.96.8.c.1 | $92$ | $24$ | $24$ | $8$ |
| 104.8.0.a.1 | $104$ | $2$ | $2$ | $0$ |
| 104.8.0.b.1 | $104$ | $2$ | $2$ | $0$ |
| 116.8.0.a.1 | $116$ | $2$ | $2$ | $0$ |
| 116.8.0.b.1 | $116$ | $2$ | $2$ | $0$ |
| 116.120.9.g.1 | $116$ | $30$ | $30$ | $9$ |
| 120.8.0.e.1 | $120$ | $2$ | $2$ | $0$ |
| 120.8.0.f.1 | $120$ | $2$ | $2$ | $0$ |
| 124.8.0.a.1 | $124$ | $2$ | $2$ | $0$ |
| 124.8.0.b.1 | $124$ | $2$ | $2$ | $0$ |
| 124.128.10.a.1 | $124$ | $32$ | $32$ | $10$ |
| 132.8.0.c.1 | $132$ | $2$ | $2$ | $0$ |
| 132.8.0.d.1 | $132$ | $2$ | $2$ | $0$ |
| 136.8.0.a.1 | $136$ | $2$ | $2$ | $0$ |
| 136.8.0.b.1 | $136$ | $2$ | $2$ | $0$ |
| 140.8.0.a.1 | $140$ | $2$ | $2$ | $0$ |
| 140.8.0.b.1 | $140$ | $2$ | $2$ | $0$ |
| 148.8.0.a.1 | $148$ | $2$ | $2$ | $0$ |
| 148.8.0.b.1 | $148$ | $2$ | $2$ | $0$ |
| 148.152.11.a.1 | $148$ | $38$ | $38$ | $11$ |
| 152.8.0.a.1 | $152$ | $2$ | $2$ | $0$ |
| 152.8.0.b.1 | $152$ | $2$ | $2$ | $0$ |
| 156.8.0.c.1 | $156$ | $2$ | $2$ | $0$ |
| 156.8.0.d.1 | $156$ | $2$ | $2$ | $0$ |
| 164.8.0.a.1 | $164$ | $2$ | $2$ | $0$ |
| 164.8.0.b.1 | $164$ | $2$ | $2$ | $0$ |
| 164.168.13.g.1 | $164$ | $42$ | $42$ | $13$ |
| 168.8.0.e.1 | $168$ | $2$ | $2$ | $0$ |
| 168.8.0.f.1 | $168$ | $2$ | $2$ | $0$ |
| 172.8.0.a.1 | $172$ | $2$ | $2$ | $0$ |
| 172.8.0.b.1 | $172$ | $2$ | $2$ | $0$ |
| 172.176.14.a.1 | $172$ | $44$ | $44$ | $14$ |
| 184.8.0.a.1 | $184$ | $2$ | $2$ | $0$ |
| 184.8.0.b.1 | $184$ | $2$ | $2$ | $0$ |
| 188.8.0.a.1 | $188$ | $2$ | $2$ | $0$ |
| 188.8.0.b.1 | $188$ | $2$ | $2$ | $0$ |
| 188.192.16.c.1 | $188$ | $48$ | $48$ | $16$ |
| 204.8.0.c.1 | $204$ | $2$ | $2$ | $0$ |
| 204.8.0.d.1 | $204$ | $2$ | $2$ | $0$ |
| 212.8.0.a.1 | $212$ | $2$ | $2$ | $0$ |
| 212.8.0.b.1 | $212$ | $2$ | $2$ | $0$ |
| 212.216.17.g.1 | $212$ | $54$ | $54$ | $17$ |
| 220.8.0.a.1 | $220$ | $2$ | $2$ | $0$ |
| 220.8.0.b.1 | $220$ | $2$ | $2$ | $0$ |
| 228.8.0.c.1 | $228$ | $2$ | $2$ | $0$ |
| 228.8.0.d.1 | $228$ | $2$ | $2$ | $0$ |
| 232.8.0.a.1 | $232$ | $2$ | $2$ | $0$ |
| 232.8.0.b.1 | $232$ | $2$ | $2$ | $0$ |
| 236.8.0.a.1 | $236$ | $2$ | $2$ | $0$ |
| 236.8.0.b.1 | $236$ | $2$ | $2$ | $0$ |
| 236.240.20.c.1 | $236$ | $60$ | $60$ | $20$ |
| 244.8.0.a.1 | $244$ | $2$ | $2$ | $0$ |
| 244.8.0.b.1 | $244$ | $2$ | $2$ | $0$ |
| 244.248.19.a.1 | $244$ | $62$ | $62$ | $19$ |
| 248.8.0.a.1 | $248$ | $2$ | $2$ | $0$ |
| 248.8.0.b.1 | $248$ | $2$ | $2$ | $0$ |
| 260.8.0.a.1 | $260$ | $2$ | $2$ | $0$ |
| 260.8.0.b.1 | $260$ | $2$ | $2$ | $0$ |
| 264.8.0.e.1 | $264$ | $2$ | $2$ | $0$ |
| 264.8.0.f.1 | $264$ | $2$ | $2$ | $0$ |
| 268.8.0.a.1 | $268$ | $2$ | $2$ | $0$ |
| 268.8.0.b.1 | $268$ | $2$ | $2$ | $0$ |
| 268.272.22.a.1 | $268$ | $68$ | $68$ | $22$ |
| 276.8.0.c.1 | $276$ | $2$ | $2$ | $0$ |
| 276.8.0.d.1 | $276$ | $2$ | $2$ | $0$ |
| 280.8.0.a.1 | $280$ | $2$ | $2$ | $0$ |
| 280.8.0.b.1 | $280$ | $2$ | $2$ | $0$ |
| 284.8.0.a.1 | $284$ | $2$ | $2$ | $0$ |
| 284.8.0.b.1 | $284$ | $2$ | $2$ | $0$ |
| 284.288.24.c.1 | $284$ | $72$ | $72$ | $24$ |
| 292.8.0.a.1 | $292$ | $2$ | $2$ | $0$ |
| 292.8.0.b.1 | $292$ | $2$ | $2$ | $0$ |
| 292.296.23.a.1 | $292$ | $74$ | $74$ | $23$ |
| 296.8.0.a.1 | $296$ | $2$ | $2$ | $0$ |
| 296.8.0.b.1 | $296$ | $2$ | $2$ | $0$ |
| 308.8.0.a.1 | $308$ | $2$ | $2$ | $0$ |
| 308.8.0.b.1 | $308$ | $2$ | $2$ | $0$ |
| 312.8.0.e.1 | $312$ | $2$ | $2$ | $0$ |
| 312.8.0.f.1 | $312$ | $2$ | $2$ | $0$ |
| 316.8.0.a.1 | $316$ | $2$ | $2$ | $0$ |
| 316.8.0.b.1 | $316$ | $2$ | $2$ | $0$ |
| 328.8.0.a.1 | $328$ | $2$ | $2$ | $0$ |
| 328.8.0.b.1 | $328$ | $2$ | $2$ | $0$ |
| 332.8.0.a.1 | $332$ | $2$ | $2$ | $0$ |
| 332.8.0.b.1 | $332$ | $2$ | $2$ | $0$ |