Elliptic curves over $\Q$ of conductor 1089

Conductor	j-invariant	Rank	Torsion	Complex multiplication
Bad$\ p$	Discriminant	Regulator	Analytic order of Ш	Galois image
Isogeny class size	Isogeny class degree	Cyclic isogeny degree	$p\ $div$\ $|Ш|	Nonmax$\ \ell$
Curves per isogeny class	Reduction	Integral points	Faltings height

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Results (23 matches)

 

Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Semistable	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	Weierstrass coefficients	Weierstrass equation
1089.a1	1089k1	1089.a	1089k	$1$	$1$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{9} \cdot 11^{2} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.177816633$	$1$		$8$	$288$	$-0.087104$	$45056/27$	$[0, 0, 1, 33, -14]$	\(y^2+y=x^3+33x-14\)
1089.b1	1089j3	1089.b	1089j	$3$	$25$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{7} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	25.60.0.2	5B.4.2	$7.655751664$	$1$		$0$	$18000$	$2.244965$	$-52893159101157376/11$	$[0, 0, 1, -8516343, -9565943918]$	\(y^2+y=x^3-8516343x-9565943918\)
1089.b2	1089j2	1089.b	1089j	$3$	$25$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{11} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	5.60.0.1	5Cs.4.1	$1.531150332$	$1$		$0$	$3600$	$1.440245$	$-122023936/161051$	$[0, 0, 1, -11253, -832208]$	\(y^2+y=x^3-11253x-832208\)
1089.b3	1089j1	1089.b	1089j	$3$	$25$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{7} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$5$	25.60.0.1	5B.4.1	$0.306230066$	$1$		$6$	$720$	$0.635525$	$-4096/11$	$[0, 0, 1, -363, 6322]$	\(y^2+y=x^3-363x+6322\)
1089.c1	1089h2	1089.c	1089h	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{8} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		✓	$2, 11$	4.2.0.1, 11.60.1.3	11B.10.5	$1.460386125$	$1$		$4$	$1584$	$1.230293$	$-24729001$	$[1, -1, 1, -32693, -2267130]$	\(y^2+xy+y=x^3-x^2-32693x-2267130\)
1089.c2	1089h1	1089.c	1089h	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{4} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		✓	$2, 11$	4.2.0.1, 11.60.1.4	11B.10.4	$0.132762375$	$1$		$6$	$144$	$0.031345$	$-121$	$[1, -1, 1, -23, 168]$	\(y^2+xy+y=x^3-x^2-23x+168\)
1089.d1	1089d2	1089.d	1089d	$2$	$2$	\( 3^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 11^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	2.3.0.1	2B	$1$	$1$		$0$	$2880$	$1.259563$	$19034163/121$	$[1, -1, 1, -18173, 942274]$	\(y^2+xy+y=x^3-x^2-18173x+942274\)
1089.d2	1089d1	1089.d	1089d	$2$	$2$	\( 3^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 11^{7} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	2.3.0.1	2B	$1$	$1$		$1$	$1440$	$0.912990$	$19683/11$	$[1, -1, 1, -1838, -5156]$	\(y^2+xy+y=x^3-x^2-1838x-5156\)
1089.e1	1089b1	1089.e	1089b	$2$	$3$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{3} \cdot 11^{10} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$11$	11.165.5.1	11Nn.1.4	$1$	$1$		$0$	$1232$	$0.951781$	$0$	$[0, 0, 1, 0, -40263]$	\(y^2+y=x^3-40263\)
1089.e2	1089b2	1089.e	1089b	$2$	$3$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{9} \cdot 11^{10} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$11$	11.165.5.1	11Nn.1.4	$1$	$1$		$0$	$3696$	$1.501087$	$0$	$[0, 0, 1, 0, 1087094]$	\(y^2+y=x^3+1087094\)
1089.f1	1089a2	1089.f	1089a	$2$	$3$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{9} \cdot 11^{4} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3, 11$	27.648.18.4, 11.165.5.1	3B.1.2, 11Nn.1.4	$1$	$1$		$0$	$336$	$0.302140$	$0$	$[0, 0, 1, 0, -817]$	\(y^2+y=x^3-817\)
1089.f2	1089a1	1089.f	1089a	$2$	$3$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{3} \cdot 11^{4} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3, 11$	27.648.18.1, 11.165.5.1	3B.1.1, 11Nn.1.4	$1$	$1$		$2$	$112$	$-0.247166$	$0$	$[0, 0, 1, 0, 30]$	\(y^2+y=x^3+30\)
1089.g1	1089e2	1089.g	1089e	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{9} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-11})$	$-11$	$N(\mathrm{U}(1))$		✓				$1$	$1$		$0$	$1320$	$1.125181$	$-32768$	$[0, 0, 1, -7986, 281839]$	\(y^2+y=x^3-7986x+281839\)
1089.g2	1089e1	1089.g	1089e	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{3} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-11})$	$-11$	$N(\mathrm{U}(1))$		✓				$1$	$1$		$0$	$120$	$-0.073767$	$-32768$	$[0, 0, 1, -66, -212]$	\(y^2+y=x^3-66x-212\)
1089.h1	1089c2	1089.h	1089c	$2$	$2$	\( 3^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 11^{8} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	2.3.0.1	2B	$1$	$1$		$0$	$960$	$0.710258$	$19034163/121$	$[1, -1, 0, -2019, -34226]$	\(y^2+xy=x^3-x^2-2019x-34226\)
1089.h2	1089c1	1089.h	1089c	$2$	$2$	\( 3^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 11^{7} \)	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	2.3.0.1	2B	$1$	$1$		$1$	$480$	$0.363684$	$19683/11$	$[1, -1, 0, -204, 259]$	\(y^2+xy=x^3-x^2-204x+259\)
1089.i1	1089f2	1089.i	1089f	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{10} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		✓	$2, 11$	4.2.0.1, 11.60.1.4	11B.10.4	$5.138097620$	$1$		$2$	$1584$	$1.230293$	$-121$	$[1, -1, 0, -2745, -215726]$	\(y^2+xy=x^3-x^2-2745x-215726\)
1089.i2	1089f1	1089.i	1089f	$2$	$11$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{6} \cdot 11^{2} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		✓	$2, 11$	4.2.0.1, 11.60.1.3	11B.10.5	$0.467099783$	$1$		$2$	$144$	$0.031345$	$-24729001$	$[1, -1, 0, -270, 1777]$	\(y^2+xy=x^3-x^2-270x+1777\)
1089.j1	1089g3	1089.j	1089g	$4$	$4$	\( 3^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 11^{10} \)	$1$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.12.0.14	2B	$2.957701004$	$1$		$2$	$5760$	$1.735622$	$347873904937/395307$	$[1, -1, 0, -159561, 24548134]$	\(y^2+xy=x^3-x^2-159561x+24548134\)
1089.j2	1089g2	1089.j	1089g	$4$	$4$	\( 3^{2} \cdot 11^{2} \)	\( 3^{12} \cdot 11^{8} \)	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	4.12.0.2	2Cs	$5.915402008$	$1$		$2$	$2880$	$1.389048$	$169112377/88209$	$[1, -1, 0, -12546, 173047]$	\(y^2+xy=x^3-x^2-12546x+173047\)
1089.j3	1089g1	1089.j	1089g	$4$	$4$	\( 3^{2} \cdot 11^{2} \)	\( 3^{9} \cdot 11^{7} \)	$1$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.12.0.14	2B	$2.957701004$	$1$		$3$	$1440$	$1.042475$	$30664297/297$	$[1, -1, 0, -7101, -226616]$	\(y^2+xy=x^3-x^2-7101x-226616\)
1089.j4	1089g4	1089.j	1089g	$4$	$4$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{18} \cdot 11^{7} \)	$1$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$			$2$	8.12.0.14	2B	$2.957701004$	$1$		$0$	$5760$	$1.735622$	$9090072503/5845851$	$[1, -1, 0, 47349, 1311052]$	\(y^2+xy=x^3-x^2+47349x+1311052\)
1089.k1	1089i1	1089.k	1089i	$1$	$1$	\( 3^{2} \cdot 11^{2} \)	\( - 3^{9} \cdot 11^{8} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$						$0.677917086$	$1$		$0$	$3168$	$1.111843$	$45056/27$	$[0, 0, 1, 3993, 18301]$	\(y^2+y=x^3+3993x+18301\)