Elliptic curves over \(\Q(\sqrt{-165}) \) of conductor 55.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100 over imaginary quadratic fields with absolute discriminant 660

Note: The completeness Only modular elliptic curves are included

Results (32 matches)

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Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
55.1-a1	55.1-a	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{14} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$2.384490492$	$14.16497688$	2.629477707	\( \frac{59319}{55} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( 20\) , \( 22\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2+20{x}+22$
55.1-a2	55.1-a	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{16} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$1.192245246$	$7.082488443$	2.629477707	\( \frac{8120601}{3025} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -105\) , \( 272\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-105{x}+272$
55.1-a3	55.1-a	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{14} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$0.596122623$	$3.541244221$	2.629477707	\( \frac{2749884201}{73205} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -730\) , \( -7228\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-730{x}-7228$
55.1-a4	55.1-a	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{20} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$0.596122623$	$3.541244221$	2.629477707	\( \frac{22930509321}{6875} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -1480\) , \( 22272\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-1480{x}+22272$
55.1-b1	55.1-b	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{2} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$3.452752300$	$14.16497688$	7.614989644	\( \frac{59319}{55} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( 7\) , \( -8\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2+7{x}-8$
55.1-b2	55.1-b	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{4} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{3} \)	$3.452752300$	$7.082488443$	7.614989644	\( \frac{8120601}{3025} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -38\) , \( -44\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-38{x}-44$
55.1-b3	55.1-b	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{2} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$0.863188075$	$3.541244221$	7.614989644	\( \frac{2749884201}{73205} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -263\) , \( 1666\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-263{x}+1666$
55.1-b4	55.1-b	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{8} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$3.452752300$	$3.541244221$	7.614989644	\( \frac{22930509321}{6875} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -533\) , \( -4598\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-533{x}-4598$
55.1-c1	55.1-c	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{14} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$15.68344211$	$14.16497688$	8.647394808	\( \frac{59319}{55} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( 98\) , \( -316\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2+98{x}-316$
55.1-c2	55.1-c	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{4} \cdot 11^{16} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{3} \)	$3.920860527$	$7.082488443$	8.647394808	\( \frac{8120601}{3025} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -507\) , \( -2494\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-507{x}-2494$
55.1-c3	55.1-c	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{20} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$3.920860527$	$3.541244221$	8.647394808	\( \frac{2749884201}{73205} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -3532\) , \( 79786\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-3532{x}+79786$
55.1-c4	55.1-c	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{8} \cdot 11^{14} \)	$6.25174$	$(5,a), (11,a)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{4} \)	$3.920860527$	$3.541244221$	8.647394808	\( \frac{22930509321}{6875} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -7162\) , \( -231426\bigr] \)	${y}^2+{x}{y}+{y}={x}^3-{x}^2-7162{x}-231426$
55.1-d1	55.1-d	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$5.189627184$	$14.16497688$	5.722819627	\( \frac{59319}{55} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 568\) , \( -2611\bigr] \)	${y}^2+a{x}{y}={x}^3+568{x}-2611$
55.1-d2	55.1-d	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{4} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$2.594813592$	$7.082488443$	5.722819627	\( \frac{8120601}{3025} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 563\) , \( -2544\bigr] \)	${y}^2+a{x}{y}={x}^3+563{x}-2544$
55.1-d3	55.1-d	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$5.189627184$	$3.541244221$	5.722819627	\( \frac{2749884201}{73205} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 538\) , \( -2139\bigr] \)	${y}^2+a{x}{y}={x}^3+538{x}-2139$
55.1-d4	55.1-d	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{8} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$1.297406796$	$3.541244221$	5.722819627	\( \frac{22930509321}{6875} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 508\) , \( -1961\bigr] \)	${y}^2+a{x}{y}={x}^3+508{x}-1961$
55.1-e1	55.1-e	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{2} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 3$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1.664591700$	$14.16497688$	7.342460423	\( \frac{59319}{55} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 657\) , \( -2653\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+657{x}-2653$
55.1-e2	55.1-e	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{4} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 3$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$64$	\( 2^{2} \)	$0.832295850$	$7.082488443$	7.342460423	\( \frac{8120601}{3025} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 612\) , \( -1987\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+612{x}-1987$
55.1-e3	55.1-e	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{2} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 3$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$1.664591700$	$3.541244221$	7.342460423	\( \frac{2749884201}{73205} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 387\) , \( -547\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+387{x}-547$
55.1-e4	55.1-e	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 3^{12} \cdot 5^{8} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 3$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$1.664591700$	$3.541244221$	7.342460423	\( \frac{22930509321}{6875} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 117\) , \( 9497\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+117{x}+9497$
55.1-f1	55.1-f	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{14} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs		\( 2^{2} \)		$14.16497688$	15.28985518	\( \frac{59319}{55} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 670\) , \( -2865\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+670{x}-2865$
55.1-f2	55.1-f	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{16} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 2$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs		\( 2^{3} \)		$7.082488443$	15.28985518	\( \frac{8120601}{3025} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 545\) , \( -1365\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+545{x}-1365$
55.1-f3	55.1-f	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{14} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 2$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs		\( 2^{4} \)		$3.541244221$	15.28985518	\( \frac{2749884201}{73205} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( -80\) , \( 14885\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-80{x}+14885$
55.1-f4	55.1-f	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{20} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	$1 \le r \le 2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs		\( 2^{2} \)		$3.541244221$	15.28985518	\( \frac{22930509321}{6875} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( -830\) , \( -4115\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-830{x}-4115$
55.1-g1	55.1-g	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1$	$14.16497688$	2.205483910	\( \frac{59319}{55} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2+{x}$
55.1-g2	55.1-g	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{4} \cdot 11^{4} \)	$6.25174$	$(5,a), (11,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{3} \)	$1$	$7.082488443$	2.205483910	\( \frac{8120601}{3025} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -4\) , \( 3\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-4{x}+3$
55.1-g3	55.1-g	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{8} \)	$6.25174$	$(5,a), (11,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{2} \)	$1$	$3.541244221$	2.205483910	\( \frac{2749884201}{73205} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -29\) , \( -52\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-29{x}-52$
55.1-g4	55.1-g	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{8} \cdot 11^{2} \)	$6.25174$	$(5,a), (11,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$16$	\( 2^{4} \)	$1$	$3.541244221$	2.205483910	\( \frac{22930509321}{6875} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -59\) , \( 190\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-59{x}+190$
55.1-h1	55.1-h	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{14} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$4.314990037$	$14.16497688$	4.758320550	\( \frac{59319}{55} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 748\) , \( -3619\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+748{x}-3619$
55.1-h2	55.1-h	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{4} \cdot 11^{16} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$2.157495018$	$7.082488443$	4.758320550	\( \frac{8120601}{3025} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( 143\) , \( 7029\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+143{x}+7029$
55.1-h3	55.1-h	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{2} \cdot 11^{20} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$1.078747509$	$3.541244221$	4.758320550	\( \frac{2749884201}{73205} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( -2882\) , \( -32901\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-2882{x}-32901$
55.1-h4	55.1-h	$4$	$4$	\(\Q(\sqrt{-165}) \)	$2$	$[0, 1]$	55.1	\( 5 \cdot 11 \)	\( 5^{8} \cdot 11^{14} \)	$6.25174$	$(5,a), (11,a)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$4.314990037$	$3.541244221$	4.758320550	\( \frac{22930509321}{6875} \)	\( \bigl[a\) , \( 0\) , \( a\) , \( -6512\) , \( 329131\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-6512{x}+329131$

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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.