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Properties

Label 2.6.0.a.1
Level $2$
Index $6$
Genus $0$
Analytic rank $0$
Cusps $3$
$\Q$-cusps $3$

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This modular curve parameterizes elliptic curves which are quadratic twists of curves in Legendre form.

Invariants

Level: $2$ $\SL_2$-level: $2$
Index: $6$ $\PSL_2$-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
Sutherland and Zywina (SZ) label: 2C0-2a
Rouse and Zureick-Brown (RZB) label: X8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 2.6.0.1
Sutherland (S) label: 2Cs

Level structure

$\GL_2(\Z/2\Z)$-generators: trivial subgroup
$\GL_2(\Z/2\Z)$-subgroup: $C_1$
Contains $-I$: yes
Quadratic refinements: 4.12.0-2.a.1.1, 4.12.0-2.a.1.2, 8.12.0-2.a.1.1, 8.12.0-2.a.1.2, 12.12.0-2.a.1.1, 12.12.0-2.a.1.2, 20.12.0-2.a.1.1, 20.12.0-2.a.1.2, 24.12.0-2.a.1.1, 24.12.0-2.a.1.2, 28.12.0-2.a.1.1, 28.12.0-2.a.1.2, 40.12.0-2.a.1.1, 40.12.0-2.a.1.2, 44.12.0-2.a.1.1, 44.12.0-2.a.1.2, 52.12.0-2.a.1.1, 52.12.0-2.a.1.2, 56.12.0-2.a.1.1, 56.12.0-2.a.1.2, 60.12.0-2.a.1.1, 60.12.0-2.a.1.2, 68.12.0-2.a.1.1, 68.12.0-2.a.1.2, 76.12.0-2.a.1.1, 76.12.0-2.a.1.2, 84.12.0-2.a.1.1, 84.12.0-2.a.1.2, 88.12.0-2.a.1.1, 88.12.0-2.a.1.2, 92.12.0-2.a.1.1, 92.12.0-2.a.1.2, 104.12.0-2.a.1.1, 104.12.0-2.a.1.2, 116.12.0-2.a.1.1, 116.12.0-2.a.1.2, 120.12.0-2.a.1.1, 120.12.0-2.a.1.2, 124.12.0-2.a.1.1, 124.12.0-2.a.1.2, 132.12.0-2.a.1.1, 132.12.0-2.a.1.2, 136.12.0-2.a.1.1, 136.12.0-2.a.1.2, 140.12.0-2.a.1.1, 140.12.0-2.a.1.2, 148.12.0-2.a.1.1, 148.12.0-2.a.1.2, 152.12.0-2.a.1.1, 152.12.0-2.a.1.2, 156.12.0-2.a.1.1, 156.12.0-2.a.1.2, 164.12.0-2.a.1.1, 164.12.0-2.a.1.2, 168.12.0-2.a.1.1, 168.12.0-2.a.1.2, 172.12.0-2.a.1.1, 172.12.0-2.a.1.2, 184.12.0-2.a.1.1, 184.12.0-2.a.1.2, 188.12.0-2.a.1.1, 188.12.0-2.a.1.2, 204.12.0-2.a.1.1, 204.12.0-2.a.1.2, 212.12.0-2.a.1.1, 212.12.0-2.a.1.2, 220.12.0-2.a.1.1, 220.12.0-2.a.1.2, 228.12.0-2.a.1.1, 228.12.0-2.a.1.2, 232.12.0-2.a.1.1, 232.12.0-2.a.1.2, 236.12.0-2.a.1.1, 236.12.0-2.a.1.2, 244.12.0-2.a.1.1, 244.12.0-2.a.1.2, 248.12.0-2.a.1.1, 248.12.0-2.a.1.2, 260.12.0-2.a.1.1, 260.12.0-2.a.1.2, 264.12.0-2.a.1.1, 264.12.0-2.a.1.2, 268.12.0-2.a.1.1, 268.12.0-2.a.1.2, 276.12.0-2.a.1.1, 276.12.0-2.a.1.2, 280.12.0-2.a.1.1, 280.12.0-2.a.1.2, 284.12.0-2.a.1.1, 284.12.0-2.a.1.2, 292.12.0-2.a.1.1, 292.12.0-2.a.1.2, 296.12.0-2.a.1.1, 296.12.0-2.a.1.2, 308.12.0-2.a.1.1, 308.12.0-2.a.1.2, 312.12.0-2.a.1.1, 312.12.0-2.a.1.2, 316.12.0-2.a.1.1, 316.12.0-2.a.1.2, 328.12.0-2.a.1.1, 328.12.0-2.a.1.2, 332.12.0-2.a.1.1, 332.12.0-2.a.1.2
Cyclic 2-isogeny field degree: $1$
Cyclic 2-torsion field degree: $1$
Full 2-torsion field degree: $1$

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

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_{\mathrm{ns}}(2)$ $2$ $3$ $3$ $0$ $0$
$X_0(2)$ $2$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
4.12.0.a.1 $4$ $2$ $2$ $0$
$X_{\pm1}(2,4)$ $4$ $2$ $2$ $0$
6.18.1.a.1 $6$ $3$ $3$ $1$
$X_{\pm1}(2,6)$ $6$ $4$ $4$ $0$
8.12.0.a.1 $8$ $2$ $2$ $0$
8.12.0.b.1 $8$ $2$ $2$ $0$
10.30.2.a.1 $10$ $5$ $5$ $2$
10.36.1.a.1 $10$ $6$ $6$ $1$
10.60.3.a.1 $10$ $10$ $10$ $3$
12.12.0.a.1 $12$ $2$ $2$ $0$
12.12.0.b.1 $12$ $2$ $2$ $0$
14.48.2.a.1 $14$ $8$ $8$ $2$
14.126.7.a.1 $14$ $21$ $21$ $7$
14.168.9.a.1 $14$ $28$ $28$ $9$
18.162.10.a.1 $18$ $27$ $27$ $10$
20.12.0.a.1 $20$ $2$ $2$ $0$
20.12.0.b.1 $20$ $2$ $2$ $0$
22.72.4.a.1 $22$ $12$ $12$ $4$
22.330.21.a.1 $22$ $55$ $55$ $21$
22.330.21.b.1 $22$ $55$ $55$ $21$
22.396.25.a.1 $22$ $66$ $66$ $25$
24.12.0.a.1 $24$ $2$ $2$ $0$
24.12.0.b.1 $24$ $2$ $2$ $0$
26.84.5.a.1 $26$ $14$ $14$ $5$
26.468.31.a.1 $26$ $78$ $78$ $31$
26.546.36.a.1 $26$ $91$ $91$ $36$
26.546.36.b.1 $26$ $91$ $91$ $36$
28.12.0.a.1 $28$ $2$ $2$ $0$
28.12.0.b.1 $28$ $2$ $2$ $0$
34.108.7.a.1 $34$ $18$ $18$ $7$
34.816.57.a.1 $34$ $136$ $136$ $57$
34.918.64.a.1 $34$ $153$ $153$ $64$
38.120.8.a.1 $38$ $20$ $20$ $8$
38.1026.73.a.1 $38$ $171$ $171$ $73$
38.1140.81.a.1 $38$ $190$ $190$ $81$
38.1710.121.a.1 $38$ $285$ $285$ $121$
40.12.0.a.1 $40$ $2$ $2$ $0$
40.12.0.b.1 $40$ $2$ $2$ $0$
44.12.0.a.1 $44$ $2$ $2$ $0$
44.12.0.b.1 $44$ $2$ $2$ $0$
46.144.10.a.1 $46$ $24$ $24$ $10$
46.1518.111.a.1 $46$ $253$ $253$ $111$
46.1656.121.a.1 $46$ $276$ $276$ $121$
52.12.0.a.1 $52$ $2$ $2$ $0$
52.12.0.b.1 $52$ $2$ $2$ $0$
56.12.0.a.1 $56$ $2$ $2$ $0$
56.12.0.b.1 $56$ $2$ $2$ $0$
58.180.13.a.1 $58$ $30$ $30$ $13$
58.2436.183.a.1 $58$ $406$ $406$ $183$
58.2610.196.a.1 $58$ $435$ $435$ $196$
58.6090.456.a.1 $58$ $1015$ $1015$ $456$
60.12.0.a.1 $60$ $2$ $2$ $0$
60.12.0.b.1 $60$ $2$ $2$ $0$
62.192.14.a.1 $62$ $32$ $32$ $14$
62.2790.211.a.1 $62$ $465$ $465$ $211$
62.2976.225.a.1 $62$ $496$ $496$ $225$
68.12.0.a.1 $68$ $2$ $2$ $0$
68.12.0.b.1 $68$ $2$ $2$ $0$
74.228.17.a.1 $74$ $38$ $38$ $17$
76.12.0.a.1 $76$ $2$ $2$ $0$
76.12.0.b.1 $76$ $2$ $2$ $0$
82.252.19.a.1 $82$ $42$ $42$ $19$
84.12.0.a.1 $84$ $2$ $2$ $0$
84.12.0.b.1 $84$ $2$ $2$ $0$
86.264.20.a.1 $86$ $44$ $44$ $20$
88.12.0.a.1 $88$ $2$ $2$ $0$
88.12.0.b.1 $88$ $2$ $2$ $0$
92.12.0.a.1 $92$ $2$ $2$ $0$
92.12.0.b.1 $92$ $2$ $2$ $0$
94.288.22.a.1 $94$ $48$ $48$ $22$
104.12.0.a.1 $104$ $2$ $2$ $0$
104.12.0.b.1 $104$ $2$ $2$ $0$
116.12.0.a.1 $116$ $2$ $2$ $0$
116.12.0.b.1 $116$ $2$ $2$ $0$
120.12.0.a.1 $120$ $2$ $2$ $0$
120.12.0.b.1 $120$ $2$ $2$ $0$
124.12.0.a.1 $124$ $2$ $2$ $0$
124.12.0.b.1 $124$ $2$ $2$ $0$
132.12.0.a.1 $132$ $2$ $2$ $0$
132.12.0.b.1 $132$ $2$ $2$ $0$
136.12.0.a.1 $136$ $2$ $2$ $0$
136.12.0.b.1 $136$ $2$ $2$ $0$
140.12.0.a.1 $140$ $2$ $2$ $0$
140.12.0.b.1 $140$ $2$ $2$ $0$
148.12.0.a.1 $148$ $2$ $2$ $0$
148.12.0.b.1 $148$ $2$ $2$ $0$
152.12.0.a.1 $152$ $2$ $2$ $0$
152.12.0.b.1 $152$ $2$ $2$ $0$
156.12.0.a.1 $156$ $2$ $2$ $0$
156.12.0.b.1 $156$ $2$ $2$ $0$
164.12.0.a.1 $164$ $2$ $2$ $0$
164.12.0.b.1 $164$ $2$ $2$ $0$
168.12.0.a.1 $168$ $2$ $2$ $0$
168.12.0.b.1 $168$ $2$ $2$ $0$
172.12.0.a.1 $172$ $2$ $2$ $0$
172.12.0.b.1 $172$ $2$ $2$ $0$
184.12.0.a.1 $184$ $2$ $2$ $0$
184.12.0.b.1 $184$ $2$ $2$ $0$
188.12.0.a.1 $188$ $2$ $2$ $0$
188.12.0.b.1 $188$ $2$ $2$ $0$
204.12.0.a.1 $204$ $2$ $2$ $0$
204.12.0.b.1 $204$ $2$ $2$ $0$
212.12.0.a.1 $212$ $2$ $2$ $0$
212.12.0.b.1 $212$ $2$ $2$ $0$
220.12.0.a.1 $220$ $2$ $2$ $0$
220.12.0.b.1 $220$ $2$ $2$ $0$
228.12.0.a.1 $228$ $2$ $2$ $0$
228.12.0.b.1 $228$ $2$ $2$ $0$
232.12.0.a.1 $232$ $2$ $2$ $0$
232.12.0.b.1 $232$ $2$ $2$ $0$
236.12.0.a.1 $236$ $2$ $2$ $0$
236.12.0.b.1 $236$ $2$ $2$ $0$
244.12.0.a.1 $244$ $2$ $2$ $0$
244.12.0.b.1 $244$ $2$ $2$ $0$
248.12.0.a.1 $248$ $2$ $2$ $0$
248.12.0.b.1 $248$ $2$ $2$ $0$
260.12.0.a.1 $260$ $2$ $2$ $0$
260.12.0.b.1 $260$ $2$ $2$ $0$
264.12.0.a.1 $264$ $2$ $2$ $0$
264.12.0.b.1 $264$ $2$ $2$ $0$
268.12.0.a.1 $268$ $2$ $2$ $0$
268.12.0.b.1 $268$ $2$ $2$ $0$
276.12.0.a.1 $276$ $2$ $2$ $0$
276.12.0.b.1 $276$ $2$ $2$ $0$
280.12.0.a.1 $280$ $2$ $2$ $0$
280.12.0.b.1 $280$ $2$ $2$ $0$
284.12.0.a.1 $284$ $2$ $2$ $0$
284.12.0.b.1 $284$ $2$ $2$ $0$
292.12.0.a.1 $292$ $2$ $2$ $0$
292.12.0.b.1 $292$ $2$ $2$ $0$
296.12.0.a.1 $296$ $2$ $2$ $0$
296.12.0.b.1 $296$ $2$ $2$ $0$
308.12.0.a.1 $308$ $2$ $2$ $0$
308.12.0.b.1 $308$ $2$ $2$ $0$
312.12.0.a.1 $312$ $2$ $2$ $0$
312.12.0.b.1 $312$ $2$ $2$ $0$
316.12.0.a.1 $316$ $2$ $2$ $0$
316.12.0.b.1 $316$ $2$ $2$ $0$
328.12.0.a.1 $328$ $2$ $2$ $0$
328.12.0.b.1 $328$ $2$ $2$ $0$
332.12.0.a.1 $332$ $2$ $2$ $0$
332.12.0.b.1 $332$ $2$ $2$ $0$