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 | $1^{2}\cdot4$ | 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 $\quad(D =$ $-4,-16$) |
Other labels
Cummins and Pauli (CP) label: | 4B0 |
Rouse and Zureick-Brown (RZB) label: | X13h |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 4.12.0.7 |
Level structure
$\GL_2(\Z/4\Z)$-generators: | $\begin{bmatrix}1&2\\0&3\end{bmatrix}$, $\begin{bmatrix}3&1\\0&3\end{bmatrix}$ |
$\GL_2(\Z/4\Z)$-subgroup: | $D_4$ |
Contains $-I$: | no $\quad$ (see 4.6.0.c.1 for the level structure with $-I$) |
Cyclic 4-isogeny field degree: | $1$ |
Cyclic 4-torsion field degree: | $1$ |
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 95098 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{6}(48x^{2}-y^{2})^{3}}{x^{10}(8x-y)(8x+y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
$X_1(2,4)$ | $4$ | $2$ | $2$ | $0$ |
4.24.0-4.d.1.1 | $4$ | $2$ | $2$ | $0$ |
8.24.0-8.d.1.2 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.k.1.2 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.m.1.1 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.m.1.8 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.n.1.2 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.n.1.12 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.o.1.1 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.o.1.8 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.p.1.1 | $8$ | $2$ | $2$ | $0$ |
8.24.0-8.p.1.8 | $8$ | $2$ | $2$ | $0$ |
12.24.0-12.g.1.2 | $12$ | $2$ | $2$ | $0$ |
12.24.0-12.h.1.2 | $12$ | $2$ | $2$ | $0$ |
12.36.1-12.c.1.2 | $12$ | $3$ | $3$ | $1$ |
12.48.0-12.g.1.1 | $12$ | $4$ | $4$ | $0$ |
20.24.0-20.g.1.2 | $20$ | $2$ | $2$ | $0$ |
20.24.0-20.h.1.2 | $20$ | $2$ | $2$ | $0$ |
20.60.2-20.c.1.2 | $20$ | $5$ | $5$ | $2$ |
20.72.1-20.c.1.8 | $20$ | $6$ | $6$ | $1$ |
20.120.3-20.c.1.5 | $20$ | $10$ | $10$ | $3$ |
24.24.0-24.s.1.2 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.v.1.2 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.y.1.13 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.z.1.8 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.z.1.9 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.ba.1.8 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.ba.1.9 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.bb.1.4 | $24$ | $2$ | $2$ | $0$ |
24.24.0-24.bb.1.13 | $24$ | $2$ | $2$ | $0$ |
28.24.0-28.g.1.2 | $28$ | $2$ | $2$ | $0$ |
28.24.0-28.h.1.2 | $28$ | $2$ | $2$ | $0$ |
28.96.2-28.c.1.4 | $28$ | $8$ | $8$ | $2$ |
28.252.7-28.c.1.5 | $28$ | $21$ | $21$ | $7$ |
28.336.9-28.c.1.2 | $28$ | $28$ | $28$ | $9$ |
36.324.10-36.d.1.1 | $36$ | $27$ | $27$ | $10$ |
40.24.0-40.s.1.3 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.v.1.3 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.y.1.2 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.y.1.15 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.z.1.4 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.z.1.13 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.ba.1.4 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.ba.1.13 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.bb.1.2 | $40$ | $2$ | $2$ | $0$ |
40.24.0-40.bb.1.15 | $40$ | $2$ | $2$ | $0$ |
44.24.0-44.g.1.2 | $44$ | $2$ | $2$ | $0$ |
44.24.0-44.h.1.2 | $44$ | $2$ | $2$ | $0$ |
44.144.4-44.c.1.5 | $44$ | $12$ | $12$ | $4$ |
44.660.21-44.d.1.4 | $44$ | $55$ | $55$ | $21$ |
44.660.21-44.f.1.1 | $44$ | $55$ | $55$ | $21$ |
44.792.25-44.c.1.8 | $44$ | $66$ | $66$ | $25$ |
52.24.0-52.g.1.2 | $52$ | $2$ | $2$ | $0$ |
52.24.0-52.h.1.2 | $52$ | $2$ | $2$ | $0$ |
52.168.5-52.c.1.4 | $52$ | $14$ | $14$ | $5$ |
52.936.31-52.c.1.1 | $52$ | $78$ | $78$ | $31$ |
52.1092.36-52.d.1.1 | $52$ | $91$ | $91$ | $36$ |
52.1092.36-52.f.1.2 | $52$ | $91$ | $91$ | $36$ |
56.24.0-56.s.1.4 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.v.1.4 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.y.1.2 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.y.1.15 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.z.1.4 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.z.1.13 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.ba.1.4 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.ba.1.13 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.bb.1.2 | $56$ | $2$ | $2$ | $0$ |
56.24.0-56.bb.1.15 | $56$ | $2$ | $2$ | $0$ |
60.24.0-60.g.1.3 | $60$ | $2$ | $2$ | $0$ |
60.24.0-60.h.1.3 | $60$ | $2$ | $2$ | $0$ |
68.24.0-68.g.1.2 | $68$ | $2$ | $2$ | $0$ |
68.24.0-68.h.1.2 | $68$ | $2$ | $2$ | $0$ |
68.216.7-68.c.1.8 | $68$ | $18$ | $18$ | $7$ |
68.1632.57-68.c.1.1 | $68$ | $136$ | $136$ | $57$ |
68.1836.64-68.c.1.1 | $68$ | $153$ | $153$ | $64$ |
76.24.0-76.g.1.2 | $76$ | $2$ | $2$ | $0$ |
76.24.0-76.h.1.2 | $76$ | $2$ | $2$ | $0$ |
76.240.8-76.c.1.1 | $76$ | $20$ | $20$ | $8$ |
84.24.0-84.g.1.2 | $84$ | $2$ | $2$ | $0$ |
84.24.0-84.h.1.4 | $84$ | $2$ | $2$ | $0$ |
88.24.0-88.s.1.3 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.v.1.3 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.y.1.2 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.y.1.15 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.z.1.4 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.z.1.13 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.ba.1.4 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.ba.1.13 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.bb.1.2 | $88$ | $2$ | $2$ | $0$ |
88.24.0-88.bb.1.15 | $88$ | $2$ | $2$ | $0$ |
92.24.0-92.g.1.2 | $92$ | $2$ | $2$ | $0$ |
92.24.0-92.h.1.2 | $92$ | $2$ | $2$ | $0$ |
92.288.10-92.c.1.2 | $92$ | $24$ | $24$ | $10$ |
104.24.0-104.s.1.3 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.v.1.3 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.y.1.2 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.y.1.15 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.z.1.4 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.z.1.13 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.ba.1.4 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.ba.1.13 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.bb.1.2 | $104$ | $2$ | $2$ | $0$ |
104.24.0-104.bb.1.15 | $104$ | $2$ | $2$ | $0$ |
116.24.0-116.g.1.2 | $116$ | $2$ | $2$ | $0$ |
116.24.0-116.h.1.2 | $116$ | $2$ | $2$ | $0$ |
116.360.13-116.c.1.3 | $116$ | $30$ | $30$ | $13$ |
120.24.0-120.s.1.3 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.v.1.3 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.y.1.4 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.y.1.29 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.z.1.8 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.z.1.25 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.ba.1.8 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.ba.1.25 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.bb.1.4 | $120$ | $2$ | $2$ | $0$ |
120.24.0-120.bb.1.29 | $120$ | $2$ | $2$ | $0$ |
124.24.0-124.g.1.2 | $124$ | $2$ | $2$ | $0$ |
124.24.0-124.h.1.2 | $124$ | $2$ | $2$ | $0$ |
124.384.14-124.c.1.5 | $124$ | $32$ | $32$ | $14$ |
132.24.0-132.g.1.3 | $132$ | $2$ | $2$ | $0$ |
132.24.0-132.h.1.4 | $132$ | $2$ | $2$ | $0$ |
136.24.0-136.s.1.4 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.v.1.4 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.y.1.4 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.y.1.13 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.z.1.2 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.z.1.15 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.ba.1.2 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.ba.1.15 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.bb.1.4 | $136$ | $2$ | $2$ | $0$ |
136.24.0-136.bb.1.13 | $136$ | $2$ | $2$ | $0$ |
140.24.0-140.g.1.2 | $140$ | $2$ | $2$ | $0$ |
140.24.0-140.h.1.2 | $140$ | $2$ | $2$ | $0$ |
148.24.0-148.g.1.2 | $148$ | $2$ | $2$ | $0$ |
148.24.0-148.h.1.2 | $148$ | $2$ | $2$ | $0$ |
148.456.17-148.c.1.2 | $148$ | $38$ | $38$ | $17$ |
152.24.0-152.s.1.3 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.v.1.3 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.y.1.2 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.y.1.15 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.z.1.4 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.z.1.13 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.ba.1.4 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.ba.1.13 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.bb.1.2 | $152$ | $2$ | $2$ | $0$ |
152.24.0-152.bb.1.15 | $152$ | $2$ | $2$ | $0$ |
156.24.0-156.g.1.3 | $156$ | $2$ | $2$ | $0$ |
156.24.0-156.h.1.4 | $156$ | $2$ | $2$ | $0$ |
164.24.0-164.g.1.2 | $164$ | $2$ | $2$ | $0$ |
164.24.0-164.h.1.2 | $164$ | $2$ | $2$ | $0$ |
164.504.19-164.c.1.3 | $164$ | $42$ | $42$ | $19$ |
168.24.0-168.s.1.5 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.v.1.7 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.y.1.7 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.y.1.26 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.z.1.3 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.z.1.30 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.ba.1.3 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.ba.1.30 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.bb.1.7 | $168$ | $2$ | $2$ | $0$ |
168.24.0-168.bb.1.26 | $168$ | $2$ | $2$ | $0$ |
172.24.0-172.g.1.2 | $172$ | $2$ | $2$ | $0$ |
172.24.0-172.h.1.2 | $172$ | $2$ | $2$ | $0$ |
172.528.20-172.c.1.8 | $172$ | $44$ | $44$ | $20$ |
184.24.0-184.s.1.3 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.v.1.3 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.y.1.2 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.y.1.15 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.z.1.4 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.z.1.13 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.ba.1.4 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.ba.1.13 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.bb.1.2 | $184$ | $2$ | $2$ | $0$ |
184.24.0-184.bb.1.15 | $184$ | $2$ | $2$ | $0$ |
188.24.0-188.g.1.2 | $188$ | $2$ | $2$ | $0$ |
188.24.0-188.h.1.2 | $188$ | $2$ | $2$ | $0$ |
204.24.0-204.g.1.4 | $204$ | $2$ | $2$ | $0$ |
204.24.0-204.h.1.3 | $204$ | $2$ | $2$ | $0$ |
212.24.0-212.g.1.2 | $212$ | $2$ | $2$ | $0$ |
212.24.0-212.h.1.2 | $212$ | $2$ | $2$ | $0$ |
220.24.0-220.g.1.2 | $220$ | $2$ | $2$ | $0$ |
220.24.0-220.h.1.3 | $220$ | $2$ | $2$ | $0$ |
228.24.0-228.g.1.2 | $228$ | $2$ | $2$ | $0$ |
228.24.0-228.h.1.4 | $228$ | $2$ | $2$ | $0$ |
232.24.0-232.s.1.3 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.v.1.2 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.y.1.2 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.y.1.15 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.z.1.4 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.z.1.13 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.ba.1.4 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.ba.1.13 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.bb.1.2 | $232$ | $2$ | $2$ | $0$ |
232.24.0-232.bb.1.15 | $232$ | $2$ | $2$ | $0$ |
236.24.0-236.g.1.2 | $236$ | $2$ | $2$ | $0$ |
236.24.0-236.h.1.2 | $236$ | $2$ | $2$ | $0$ |
244.24.0-244.g.1.2 | $244$ | $2$ | $2$ | $0$ |
244.24.0-244.h.1.2 | $244$ | $2$ | $2$ | $0$ |
248.24.0-248.s.1.3 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.v.1.3 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.y.1.2 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.y.1.15 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.z.1.4 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.z.1.13 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.ba.1.4 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.ba.1.13 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.bb.1.2 | $248$ | $2$ | $2$ | $0$ |
248.24.0-248.bb.1.15 | $248$ | $2$ | $2$ | $0$ |
260.24.0-260.g.1.2 | $260$ | $2$ | $2$ | $0$ |
260.24.0-260.h.1.2 | $260$ | $2$ | $2$ | $0$ |
264.24.0-264.s.1.3 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.v.1.6 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.y.1.2 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.y.1.31 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.z.1.4 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.z.1.29 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.ba.1.4 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.ba.1.29 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.bb.1.2 | $264$ | $2$ | $2$ | $0$ |
264.24.0-264.bb.1.31 | $264$ | $2$ | $2$ | $0$ |
268.24.0-268.g.1.2 | $268$ | $2$ | $2$ | $0$ |
268.24.0-268.h.1.2 | $268$ | $2$ | $2$ | $0$ |
276.24.0-276.g.1.3 | $276$ | $2$ | $2$ | $0$ |
276.24.0-276.h.1.4 | $276$ | $2$ | $2$ | $0$ |
280.24.0-280.s.1.2 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.v.1.5 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.y.1.4 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.y.1.29 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.z.1.8 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.z.1.25 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.ba.1.8 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.ba.1.25 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.bb.1.4 | $280$ | $2$ | $2$ | $0$ |
280.24.0-280.bb.1.29 | $280$ | $2$ | $2$ | $0$ |
284.24.0-284.g.1.2 | $284$ | $2$ | $2$ | $0$ |
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