This curve parameterizes elliptic curves in Legendre form.
Invariants
| Level: | $4$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (all of which are rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse and Zureick-Brown (RZB) label: | X8d |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 4.12.0.1 |
Level structure
| $\GL_2(\Z/4\Z)$-generators: | $\begin{bmatrix}1&0\\2&1\end{bmatrix}$, $\begin{bmatrix}1&2\\0&1\end{bmatrix}$, $\begin{bmatrix}3&0\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/4\Z)$-subgroup: | $C_2^3$ |
| Contains $-I$: | no $\quad$ (see 2.6.0.a.1 for the level structure with $-I$) |
| Cyclic 4-isogeny field degree: | $2$ |
| Cyclic 4-torsion field degree: | $2$ |
| Full 4-torsion field degree: | $8$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 31720 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(x^{2}+192y^{2})^{3}}{y^{2}x^{6}(x-8y)^{2}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 4.24.0-4.a.1.1 | $4$ | $2$ | $2$ | $0$ |
| 4.24.0-4.a.1.2 | $4$ | $2$ | $2$ | $0$ |
| 4.24.0-4.b.1.1 | $4$ | $2$ | $2$ | $0$ |
| 4.24.0-4.b.1.2 | $4$ | $2$ | $2$ | $0$ |
| $X_1(2,4)$ | $4$ | $2$ | $2$ | $0$ |
| 8.24.0-8.a.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.a.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.a.1.3 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.b.1.1 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.b.1.2 | $8$ | $2$ | $2$ | $0$ |
| 8.24.0-8.b.1.3 | $8$ | $2$ | $2$ | $0$ |
| 12.24.0-12.a.1.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.a.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.a.1.3 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.b.1.1 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.b.1.2 | $12$ | $2$ | $2$ | $0$ |
| 12.24.0-12.b.1.3 | $12$ | $2$ | $2$ | $0$ |
| 12.36.1-6.a.1.1 | $12$ | $3$ | $3$ | $1$ |
| 12.48.0-6.a.1.1 | $12$ | $4$ | $4$ | $0$ |
| 20.24.0-20.a.1.1 | $20$ | $2$ | $2$ | $0$ |
| 20.24.0-20.a.1.2 | $20$ | $2$ | $2$ | $0$ |
| 20.24.0-20.a.1.3 | $20$ | $2$ | $2$ | $0$ |
| 20.24.0-20.b.1.1 | $20$ | $2$ | $2$ | $0$ |
| 20.24.0-20.b.1.2 | $20$ | $2$ | $2$ | $0$ |
| 20.24.0-20.b.1.3 | $20$ | $2$ | $2$ | $0$ |
| 20.60.2-10.a.1.1 | $20$ | $5$ | $5$ | $2$ |
| 20.72.1-10.a.1.1 | $20$ | $6$ | $6$ | $1$ |
| 20.120.3-10.a.1.4 | $20$ | $10$ | $10$ | $3$ |
| 24.24.0-24.a.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.a.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.a.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.b.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.b.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.24.0-24.b.1.6 | $24$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.a.1.3 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.1 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.2 | $28$ | $2$ | $2$ | $0$ |
| 28.24.0-28.b.1.3 | $28$ | $2$ | $2$ | $0$ |
| 28.96.2-14.a.1.1 | $28$ | $8$ | $8$ | $2$ |
| 28.252.7-14.a.1.1 | $28$ | $21$ | $21$ | $7$ |
| 28.336.9-14.a.1.1 | $28$ | $28$ | $28$ | $9$ |
| 36.324.10-18.a.1.1 | $36$ | $27$ | $27$ | $10$ |
| 40.24.0-40.a.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.a.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.a.1.6 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.b.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.b.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0-40.b.1.5 | $40$ | $2$ | $2$ | $0$ |
| 44.24.0-44.a.1.1 | $44$ | $2$ | $2$ | $0$ |
| 44.24.0-44.a.1.2 | $44$ | $2$ | $2$ | $0$ |
| 44.24.0-44.a.1.3 | $44$ | $2$ | $2$ | $0$ |
| 44.24.0-44.b.1.1 | $44$ | $2$ | $2$ | $0$ |
| 44.24.0-44.b.1.2 | $44$ | $2$ | $2$ | $0$ |
| 44.24.0-44.b.1.3 | $44$ | $2$ | $2$ | $0$ |
| 44.144.4-22.a.1.1 | $44$ | $12$ | $12$ | $4$ |
| 44.660.21-22.a.1.2 | $44$ | $55$ | $55$ | $21$ |
| 44.660.21-22.b.1.2 | $44$ | $55$ | $55$ | $21$ |
| 44.792.25-22.a.1.2 | $44$ | $66$ | $66$ | $25$ |
| 52.24.0-52.a.1.1 | $52$ | $2$ | $2$ | $0$ |
| 52.24.0-52.a.1.2 | $52$ | $2$ | $2$ | $0$ |
| 52.24.0-52.a.1.3 | $52$ | $2$ | $2$ | $0$ |
| 52.24.0-52.b.1.1 | $52$ | $2$ | $2$ | $0$ |
| 52.24.0-52.b.1.2 | $52$ | $2$ | $2$ | $0$ |
| 52.24.0-52.b.1.3 | $52$ | $2$ | $2$ | $0$ |
| 52.168.5-26.a.1.1 | $52$ | $14$ | $14$ | $5$ |
| 52.936.31-26.a.1.2 | $52$ | $78$ | $78$ | $31$ |
| 52.1092.36-26.a.1.1 | $52$ | $91$ | $91$ | $36$ |
| 52.1092.36-26.b.1.3 | $52$ | $91$ | $91$ | $36$ |
| 56.24.0-56.a.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.a.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.a.1.6 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.1 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.24.0-56.b.1.7 | $56$ | $2$ | $2$ | $0$ |
| 60.24.0-60.a.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.a.1.2 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.a.1.6 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.b.1.1 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.b.1.4 | $60$ | $2$ | $2$ | $0$ |
| 60.24.0-60.b.1.6 | $60$ | $2$ | $2$ | $0$ |
| 68.24.0-68.a.1.1 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.a.1.2 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.a.1.3 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.b.1.1 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.b.1.2 | $68$ | $2$ | $2$ | $0$ |
| 68.24.0-68.b.1.3 | $68$ | $2$ | $2$ | $0$ |
| 68.216.7-34.a.1.1 | $68$ | $18$ | $18$ | $7$ |
| 68.1632.57-34.a.1.2 | $68$ | $136$ | $136$ | $57$ |
| 68.1836.64-34.a.1.2 | $68$ | $153$ | $153$ | $64$ |
| 76.24.0-76.a.1.1 | $76$ | $2$ | $2$ | $0$ |
| 76.24.0-76.a.1.2 | $76$ | $2$ | $2$ | $0$ |
| 76.24.0-76.a.1.3 | $76$ | $2$ | $2$ | $0$ |
| 76.24.0-76.b.1.1 | $76$ | $2$ | $2$ | $0$ |
| 76.24.0-76.b.1.2 | $76$ | $2$ | $2$ | $0$ |
| 76.24.0-76.b.1.3 | $76$ | $2$ | $2$ | $0$ |
| 76.240.8-38.a.1.1 | $76$ | $20$ | $20$ | $8$ |
| 84.24.0-84.a.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.a.1.6 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.1 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.4 | $84$ | $2$ | $2$ | $0$ |
| 84.24.0-84.b.1.5 | $84$ | $2$ | $2$ | $0$ |
| 88.24.0-88.a.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.a.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.a.1.6 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.b.1.1 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.b.1.4 | $88$ | $2$ | $2$ | $0$ |
| 88.24.0-88.b.1.6 | $88$ | $2$ | $2$ | $0$ |
| 92.24.0-92.a.1.1 | $92$ | $2$ | $2$ | $0$ |
| 92.24.0-92.a.1.2 | $92$ | $2$ | $2$ | $0$ |
| 92.24.0-92.a.1.3 | $92$ | $2$ | $2$ | $0$ |
| 92.24.0-92.b.1.1 | $92$ | $2$ | $2$ | $0$ |
| 92.24.0-92.b.1.2 | $92$ | $2$ | $2$ | $0$ |
| 92.24.0-92.b.1.3 | $92$ | $2$ | $2$ | $0$ |
| 92.288.10-46.a.1.1 | $92$ | $24$ | $24$ | $10$ |
| 104.24.0-104.a.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.a.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.a.1.6 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.b.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.b.1.4 | $104$ | $2$ | $2$ | $0$ |
| 104.24.0-104.b.1.5 | $104$ | $2$ | $2$ | $0$ |
| 116.24.0-116.a.1.1 | $116$ | $2$ | $2$ | $0$ |
| 116.24.0-116.a.1.2 | $116$ | $2$ | $2$ | $0$ |
| 116.24.0-116.a.1.3 | $116$ | $2$ | $2$ | $0$ |
| 116.24.0-116.b.1.1 | $116$ | $2$ | $2$ | $0$ |
| 116.24.0-116.b.1.2 | $116$ | $2$ | $2$ | $0$ |
| 116.24.0-116.b.1.3 | $116$ | $2$ | $2$ | $0$ |
| 116.360.13-58.a.1.1 | $116$ | $30$ | $30$ | $13$ |
| 120.24.0-120.a.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.a.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.a.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.b.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.b.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0-120.b.1.9 | $120$ | $2$ | $2$ | $0$ |
| 124.24.0-124.a.1.1 | $124$ | $2$ | $2$ | $0$ |
| 124.24.0-124.a.1.2 | $124$ | $2$ | $2$ | $0$ |
| 124.24.0-124.a.1.3 | $124$ | $2$ | $2$ | $0$ |
| 124.24.0-124.b.1.1 | $124$ | $2$ | $2$ | $0$ |
| 124.24.0-124.b.1.2 | $124$ | $2$ | $2$ | $0$ |
| 124.24.0-124.b.1.3 | $124$ | $2$ | $2$ | $0$ |
| 124.384.14-62.a.1.1 | $124$ | $32$ | $32$ | $14$ |
| 132.24.0-132.a.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.a.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.a.1.5 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.b.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.b.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.24.0-132.b.1.5 | $132$ | $2$ | $2$ | $0$ |
| 136.24.0-136.a.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.a.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.a.1.5 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.b.1.1 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.b.1.4 | $136$ | $2$ | $2$ | $0$ |
| 136.24.0-136.b.1.5 | $136$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.4 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.a.1.8 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.1 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.2 | $140$ | $2$ | $2$ | $0$ |
| 140.24.0-140.b.1.5 | $140$ | $2$ | $2$ | $0$ |
| 148.24.0-148.a.1.1 | $148$ | $2$ | $2$ | $0$ |
| 148.24.0-148.a.1.2 | $148$ | $2$ | $2$ | $0$ |
| 148.24.0-148.a.1.3 | $148$ | $2$ | $2$ | $0$ |
| 148.24.0-148.b.1.1 | $148$ | $2$ | $2$ | $0$ |
| 148.24.0-148.b.1.2 | $148$ | $2$ | $2$ | $0$ |
| 148.24.0-148.b.1.3 | $148$ | $2$ | $2$ | $0$ |
| 148.456.17-74.a.1.1 | $148$ | $38$ | $38$ | $17$ |
| 152.24.0-152.a.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.a.1.4 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.a.1.6 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.b.1.1 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.b.1.3 | $152$ | $2$ | $2$ | $0$ |
| 152.24.0-152.b.1.6 | $152$ | $2$ | $2$ | $0$ |
| 156.24.0-156.a.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.a.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.a.1.6 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.b.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.b.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.24.0-156.b.1.6 | $156$ | $2$ | $2$ | $0$ |
| 164.24.0-164.a.1.1 | $164$ | $2$ | $2$ | $0$ |
| 164.24.0-164.a.1.2 | $164$ | $2$ | $2$ | $0$ |
| 164.24.0-164.a.1.3 | $164$ | $2$ | $2$ | $0$ |
| 164.24.0-164.b.1.1 | $164$ | $2$ | $2$ | $0$ |
| 164.24.0-164.b.1.2 | $164$ | $2$ | $2$ | $0$ |
| 164.24.0-164.b.1.3 | $164$ | $2$ | $2$ | $0$ |
| 164.504.19-82.a.1.1 | $164$ | $42$ | $42$ | $19$ |
| 168.24.0-168.a.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.a.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.24.0-168.b.1.9 | $168$ | $2$ | $2$ | $0$ |
| 172.24.0-172.a.1.1 | $172$ | $2$ | $2$ | $0$ |
| 172.24.0-172.a.1.2 | $172$ | $2$ | $2$ | $0$ |
| 172.24.0-172.a.1.3 | $172$ | $2$ | $2$ | $0$ |
| 172.24.0-172.b.1.1 | $172$ | $2$ | $2$ | $0$ |
| 172.24.0-172.b.1.2 | $172$ | $2$ | $2$ | $0$ |
| 172.24.0-172.b.1.3 | $172$ | $2$ | $2$ | $0$ |
| 172.528.20-86.a.1.1 | $172$ | $44$ | $44$ | $20$ |
| 184.24.0-184.a.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.a.1.4 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.a.1.6 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.b.1.1 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.b.1.2 | $184$ | $2$ | $2$ | $0$ |
| 184.24.0-184.b.1.7 | $184$ | $2$ | $2$ | $0$ |
| 188.24.0-188.a.1.1 | $188$ | $2$ | $2$ | $0$ |
| 188.24.0-188.a.1.2 | $188$ | $2$ | $2$ | $0$ |
| 188.24.0-188.a.1.3 | $188$ | $2$ | $2$ | $0$ |
| 188.24.0-188.b.1.1 | $188$ | $2$ | $2$ | $0$ |
| 188.24.0-188.b.1.2 | $188$ | $2$ | $2$ | $0$ |
| 188.24.0-188.b.1.3 | $188$ | $2$ | $2$ | $0$ |
| 204.24.0-204.a.1.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.a.1.2 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.a.1.5 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.b.1.1 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.b.1.4 | $204$ | $2$ | $2$ | $0$ |
| 204.24.0-204.b.1.6 | $204$ | $2$ | $2$ | $0$ |
| 212.24.0-212.a.1.1 | $212$ | $2$ | $2$ | $0$ |
| 212.24.0-212.a.1.2 | $212$ | $2$ | $2$ | $0$ |
| 212.24.0-212.a.1.3 | $212$ | $2$ | $2$ | $0$ |
| 212.24.0-212.b.1.1 | $212$ | $2$ | $2$ | $0$ |
| 212.24.0-212.b.1.2 | $212$ | $2$ | $2$ | $0$ |
| 212.24.0-212.b.1.3 | $212$ | $2$ | $2$ | $0$ |
| 220.24.0-220.a.1.1 | $220$ | $2$ | $2$ | $0$ |
| 220.24.0-220.a.1.4 | $220$ | $2$ | $2$ | $0$ |
| 220.24.0-220.a.1.5 | $220$ | $2$ | $2$ | $0$ |
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