LMFDB - Elliptic curves over $\Q$ of conductor 1690

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Conductor	Discriminant	j-invariant	Rank

Bad$\ p$	Curves per isogeny class	Complex multiplication	Torsion

Isogeny class degree	Cyclic isogeny degree	Isogeny class size	Integral points

Analytic order of Ш	$p\ $div$\ $\|Ш\|	Regulator	Reduction	Faltings height

Galois image	Adelic level	Adelic index	Adelic genus

Nonmax$\ \ell$	$abc$ quality	Szpiro ratio	Manin constant

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Results (22 matches)

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	Sato-Tate	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Adelic level	Adelic index	Adelic genus	Regulator	$Ш_{\textrm{an}}$	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
1690.a1	1690c1	1690.a	1690c	$2$	$3$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{3} \cdot 5 \cdot 13^{8} $	$1$	$\Z/3\Z$	$\Q$	$\mathrm{SU}(2)$	$3$	3.8.0.1	3B.1.1	$120$	$16$	$0$	$3.717316424$	$1$	$4$	$1872$	$0.772261$	$-658489/40$	$0.82676$	$4.57682$	$[1, 0, 1, -1694, 28056]$	$y^2+xy+y=x^3-1694x+28056$	3.8.0-3.a.1.2, 40.2.0.a.1, 120.16.0.?	$[(0, 167)]$
1690.a2	1690c2	1690.a	1690c	$2$	$3$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{9} \cdot 5^{3} \cdot 13^{8} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$3$	3.8.0.2	3B.1.2	$120$	$16$	$0$	$1.239105474$	$1$	$2$	$5616$	$1.321566$	$108750551/64000$	$1.00157$	$5.25049$	$[1, 0, 1, 9291, 45632]$	$y^2+xy+y=x^3+9291x+45632$	3.8.0-3.a.1.1, 40.2.0.a.1, 120.16.0.?	$[(14, 415)]$
1690.b1	1690a3	1690.b	1690a	$4$	$4$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{2} \cdot 5 \cdot 13^{7} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.24.0.15	2B	$1040$	$192$	$3$	$9.299534635$	$1$	$0$	$5376$	$1.478262$	$294889639316481/260$	$1.02336$	$6.55330$	$[1, -1, 0, -234350, -43607680]$	$y^2+xy=x^3-x^2-234350x-43607680$	2.3.0.a.1, 4.6.0.c.1, 8.12.0.o.1, 16.24.0.i.1, 20.12.0-4.c.1.1, $\ldots$	$[(101917/12, 20173639/12)]$
1690.b2	1690a2	1690.b	1690a	$4$	$4$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{4} \cdot 5^{2} \cdot 13^{8} $	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.24.0.35	2Cs	$520$	$192$	$3$	$4.649767317$	$1$	$2$	$2688$	$1.131689$	$72043225281/67600$	$1.01871$	$5.43428$	$[1, -1, 0, -14650, -678300]$	$y^2+xy=x^3-x^2-14650x-678300$	2.6.0.a.1, 4.12.0.a.1, 8.24.0.g.1, 20.24.0-4.a.1.2, 40.48.0-8.g.1.4, $\ldots$	$[(-281/2, 455/2)]$
1690.b3	1690a4	1690.b	1690a	$4$	$4$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{2} \cdot 5^{4} \cdot 13^{10} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.24.0.2	2B	$1040$	$192$	$3$	$2.324883658$	$1$	$2$	$5376$	$1.478262$	$-32798729601/71402500$	$1.04885$	$5.53839$	$[1, -1, 0, -11270, -1002104]$	$y^2+xy=x^3-x^2-11270x-1002104$	2.3.0.a.1, 4.24.0.c.1, 40.48.0-4.c.1.2, 52.48.0-4.c.1.1, 520.96.1.?, $\ldots$	$[(868, 24916)]$
1690.b4	1690a1	1690.b	1690a	$4$	$4$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{8} \cdot 5 \cdot 13^{7} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.24.0.15	2B	$1040$	$192$	$3$	$2.324883658$	$1$	$3$	$1344$	$0.785114$	$33076161/16640$	$0.93564$	$4.40015$	$[1, -1, 0, -1130, -5004]$	$y^2+xy=x^3-x^2-1130x-5004$	2.3.0.a.1, 4.6.0.c.1, 8.12.0.o.1, 16.24.0.i.1, 20.12.0-4.c.1.2, $\ldots$	$[(-28, 78)]$
1690.c1	1690b2	1690.c	1690b	$2$	$7$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{35} \cdot 5 \cdot 13^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$7$	7.8.0.1	7B	$3640$	$96$	$2$	$54.88869623$	$1$	$0$	$152880$	$3.034260$	$-1762712152495281/171798691840$	$1.13774$	$8.19521$	$[1, -1, 0, -13000610, -19499343980]$	$y^2+xy=x^3-x^2-13000610x-19499343980$	7.8.0.a.1, 40.2.0.a.1, 91.48.0.?, 280.16.0.?, 3640.96.2.?	$[(686347842850937155849803/992841926, 568263844856275973907944333031281963/992841926)]$
1690.c2	1690b1	1690.c	1690b	$2$	$7$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{5} \cdot 5^{7} \cdot 13^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$7$	7.8.0.1	7B	$3640$	$96$	$2$	$7.841242319$	$1$	$0$	$21840$	$2.061306$	$-2609064081/2500000$	$1.05128$	$6.49964$	$[1, -1, 0, -148160, 35808800]$	$y^2+xy=x^3-x^2-148160x+35808800$	7.8.0.a.1, 40.2.0.a.1, 91.48.0.?, 280.16.0.?, 3640.96.2.?	$[(-1133/2, 60457/2)]$
1690.d1	1690d2	1690.d	1690d	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{3} \cdot 5^{6} \cdot 13^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	2.3.0.1, 3.6.0.1	2B, 3Ns	$1560$	$144$	$5$	$1$	$1$	$0$	$864$	$0.429844$	$10260751717/125000$	$1.10093$	$4.13677$	$[1, 1, 0, -588, 5192]$	$y^2+xy=x^3+x^2-588x+5192$	2.3.0.a.1, 3.6.0.b.1, 6.18.0.b.1, 24.36.0.e.1, 39.12.0.a.1, $\ldots$	$[ ]$
1690.d2	1690d1	1690.d	1690d	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{6} \cdot 5^{3} \cdot 13^{3} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	2.3.0.1, 3.6.0.1	2B, 3Ns	$1560$	$144$	$5$	$1$	$1$	$1$	$432$	$0.083271$	$16194277/8000$	$0.94554$	$3.26876$	$[1, 1, 0, -68, -112]$	$y^2+xy=x^3+x^2-68x-112$	2.3.0.a.1, 3.6.0.b.1, 6.18.0.b.1, 24.36.0.d.1, 30.36.0.d.1, $\ldots$	$[ ]$
1690.e1	1690e1	1690.e	1690e	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{8} \cdot 5^{5} \cdot 13^{7} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.6.0.3	2B	$520$	$48$	$0$	$1$	$1$	$1$	$13440$	$1.639425$	$65787589563409/10400000$	$0.97958$	$6.35146$	$[1, 1, 0, -142132, -20681136]$	$y^2+xy=x^3+x^2-142132x-20681136$	2.3.0.a.1, 4.6.0.b.1, 40.12.0-4.b.1.4, 104.12.0.?, 130.6.0.?, $\ldots$	$[ ]$
1690.e2	1690e2	1690.e	1690e	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{4} \cdot 5^{10} \cdot 13^{8} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	4.6.0.5	2B	$520$	$48$	$0$	$1$	$1$	$0$	$26880$	$1.986000$	$-48743122863889/26406250000$	$0.98824$	$6.40001$	$[1, 1, 0, -128612, -24756064]$	$y^2+xy=x^3+x^2-128612x-24756064$	2.3.0.a.1, 4.6.0.a.1, 40.12.0-4.a.1.2, 104.12.0.?, 260.12.0.?, $\ldots$	$[ ]$
1690.f1	1690f3	1690.f	1690f	$4$	$6$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{12} \cdot 5 \cdot 13^{9} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	4.6.0.3, 3.4.0.1	2B, 3B	$1560$	$384$	$9$	$1$	$1$	$1$	$12096$	$1.512924$	$988345570681/44994560$	$0.95432$	$5.78662$	$[1, 0, 0, -35071, -2429415]$	$y^2+xy=x^3-35071x-2429415$	2.3.0.a.1, 3.4.0.a.1, 4.6.0.b.1, 6.12.0.a.1, 12.24.0.f.1, $\ldots$	$[ ]$
1690.f2	1690f1	1690.f	1690f	$4$	$6$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{4} \cdot 5^{3} \cdot 13^{7} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	4.6.0.3, 3.4.0.1	2B, 3B	$1560$	$384$	$9$	$1$	$1$	$1$	$4032$	$0.963618$	$3803721481/26000$	$0.90619$	$5.03855$	$[1, 0, 0, -5496, 155440]$	$y^2+xy=x^3-5496x+155440$	2.3.0.a.1, 3.4.0.a.1, 4.6.0.b.1, 6.12.0.a.1, 12.24.0.f.1, $\ldots$	$[ ]$
1690.f3	1690f2	1690.f	1690f	$4$	$6$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{2} \cdot 5^{6} \cdot 13^{8} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	4.6.0.5, 3.4.0.1	2B, 3B	$1560$	$384$	$9$	$1$	$1$	$0$	$8064$	$1.310192$	$-217081801/10562500$	$0.97746$	$5.25115$	$[1, 0, 0, -2116, 345396]$	$y^2+xy=x^3-2116x+345396$	2.3.0.a.1, 3.4.0.a.1, 4.6.0.a.1, 6.12.0.a.1, 12.48.0.b.1, $\ldots$	$[ ]$
1690.f4	1690f4	1690.f	1690f	$4$	$6$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{6} \cdot 5^{2} \cdot 13^{12} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	4.6.0.5, 3.4.0.1	2B, 3B	$1560$	$384$	$9$	$1$	$1$	$0$	$24192$	$1.859497$	$157376536199/7722894400$	$1.01877$	$6.13523$	$[1, 0, 0, 19009, -9232679]$	$y^2+xy=x^3+19009x-9232679$	2.3.0.a.1, 3.4.0.a.1, 4.6.0.a.1, 6.12.0.a.1, 12.48.0.b.2, $\ldots$	$[ ]$
1690.g1	1690h1	1690.g	1690h	$2$	$3$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{3} \cdot 5 \cdot 13^{2} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$3$	3.4.0.1	3B	$1560$	$16$	$0$	$0.257906098$	$1$	$4$	$144$	$-0.510215$	$-658489/40$	$0.82676$	$2.50622$	$[1, 0, 0, -10, 12]$	$y^2+xy=x^3-10x+12$	3.4.0.a.1, 39.8.0-3.a.1.1, 40.2.0.a.1, 120.8.0.?, 1560.16.0.?	$[(2, 0)]$
1690.g2	1690h2	1690.g	1690h	$2$	$3$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{9} \cdot 5^{3} \cdot 13^{2} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$3$	3.4.0.1	3B	$1560$	$16$	$0$	$0.085968699$	$1$	$10$	$432$	$0.039092$	$108750551/64000$	$1.00157$	$3.17989$	$[1, 0, 0, 55, 25]$	$y^2+xy=x^3+55x+25$	3.4.0.a.1, 39.8.0-3.a.1.2, 40.2.0.a.1, 120.8.0.?, 1560.16.0.?	$[(10, 35)]$
1690.h1	1690g2	1690.h	1690g	$2$	$7$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{35} \cdot 5 \cdot 13^{4} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$7$	7.16.0.2	7B.2.3	$3640$	$96$	$2$	$0.259597054$	$1$	$6$	$11760$	$1.751785$	$-1762712152495281/171798691840$	$1.13774$	$6.12461$	$[1, -1, 1, -76927, -8857689]$	$y^2+xy+y=x^3-x^2-76927x-8857689$	7.16.0-7.a.1.1, 40.2.0.a.1, 91.48.0.?, 280.32.0.?, 3640.96.2.?	$[(2259, 105366)]$
1690.h2	1690g1	1690.h	1690g	$2$	$7$	$ 2 \cdot 5 \cdot 13^{2} $	$ - 2^{5} \cdot 5^{7} \cdot 13^{4} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	$7$	7.16.0.1	7B.2.1	$3640$	$96$	$2$	$0.037085293$	$1$	$12$	$1680$	$0.778831$	$-2609064081/2500000$	$1.05128$	$4.42904$	$[1, -1, 1, -877, 16501]$	$y^2+xy+y=x^3-x^2-877x+16501$	7.16.0-7.a.1.2, 40.2.0.a.1, 91.48.0.?, 280.32.0.?, 3640.96.2.?	$[(-29, 144)]$
1690.i1	1690i2	1690.i	1690i	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{3} \cdot 5^{6} \cdot 13^{9} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	2.3.0.1, 3.6.0.1	2B, 3Ns	$1560$	$144$	$5$	$1$	$1$	$0$	$11232$	$1.712318$	$10260751717/125000$	$1.10093$	$6.20736$	$[1, 1, 1, -99460, 11903965]$	$y^2+xy+y=x^3+x^2-99460x+11903965$	2.3.0.a.1, 3.6.0.b.1, 6.18.0.b.1, 24.36.0.e.1, 39.12.0.a.1, $\ldots$	$[ ]$
1690.i2	1690i1	1690.i	1690i	$2$	$2$	$ 2 \cdot 5 \cdot 13^{2} $	$ 2^{6} \cdot 5^{3} \cdot 13^{9} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 3$	2.3.0.1, 3.6.0.1	2B, 3Ns	$1560$	$144$	$5$	$1$	$1$	$1$	$5616$	$1.365746$	$16194277/8000$	$0.94554$	$5.33936$	$[1, 1, 1, -11580, -188323]$	$y^2+xy+y=x^3+x^2-11580x-188323$	2.3.0.a.1, 3.6.0.b.1, 6.18.0.b.1, 24.36.0.d.1, 30.36.0.d.1, $\ldots$	$[ ]$

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Elliptic curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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