Elliptic curves over \(\Q(\sqrt{-599}) \) of conductor 36.6

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 599

Note: The completeness Only modular elliptic curves are included

Results (5 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
36.6-a1	36.6-a	$1$	$1$	\(\Q(\sqrt{-599}) \)	$2$	$[0, 1]$	36.6	\( 2^{2} \cdot 3^{2} \)	\( 2^{12} \cdot 3^{3} \cdot 83^{12} \)	$5.35707$	$(2,a), (2,a+1), (3,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 2^{3} \)	$1$	$9.782035483$	3.197465076	\( \frac{180527}{1024} a - \frac{4673801}{1024} \)	\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 1384 a - 1984\) , \( 65524 a + 464312\bigr] \)	${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(1384a-1984\right){x}+65524a+464312$
36.6-b1	36.6-b	$2$	$17$	\(\Q(\sqrt{-599}) \)	$2$	$[0, 1]$	36.6	\( 2^{2} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{6} \cdot 5^{12} \)	$5.35707$	$(2,a), (2,a+1), (3,a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$17$	17B	$1$	\( 2 \cdot 17 \)	$1.040898424$	$4.390381927$	12.69714720	\( \frac{877386405}{131072} a - \frac{1686984049}{65536} \)	\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 31 a - 475\) , \( -483 a + 825\bigr] \)	${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(31a-475\right){x}-483a+825$
36.6-b2	36.6-b	$2$	$17$	\(\Q(\sqrt{-599}) \)	$2$	$[0, 1]$	36.6	\( 2^{2} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{6} \cdot 41^{12} \)	$5.35707$	$(2,a), (2,a+1), (3,a+2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$17$	17B	$1$	\( 2 \)	$17.69527321$	$4.390381927$	12.69714720	\( -\frac{877386405}{131072} a - \frac{2496581693}{131072} \)	\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -2960 a - 5764\) , \( 194358 a - 1223148\bigr] \)	${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-2960a-5764\right){x}+194358a-1223148$
36.6-c1	36.6-c	$2$	$3$	\(\Q(\sqrt{-599}) \)	$2$	$[0, 1]$	36.6	\( 2^{2} \cdot 3^{2} \)	\( 2^{12} \cdot 3^{8} \cdot 53^{12} \)	$5.35707$	$(2,a), (2,a+1), (3,a+2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	\( 2 \cdot 3^{3} \)	$1$	$6.327947535$	1.551317561	\( \frac{797125}{4608} a + \frac{2565125}{768} \)	\( \bigl[1\) , \( -a\) , \( a\) , \( -747 a + 13597\) , \( -43584 a - 449345\bigr] \)	${y}^2+{x}{y}+a{y}={x}^3-a{x}^2+\left(-747a+13597\right){x}-43584a-449345$
36.6-c2	36.6-c	$2$	$3$	\(\Q(\sqrt{-599}) \)	$2$	$[0, 1]$	36.6	\( 2^{2} \cdot 3^{2} \)	\( 2^{4} \cdot 3^{12} \cdot 5^{12} \)	$5.35707$	$(2,a), (2,a+1), (3,a+2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	\( 2 \cdot 3 \)	$1$	$6.327947535$	1.551317561	\( \frac{4608625}{5832} a + \frac{16862375}{972} \)	\( \bigl[1\) , \( -a\) , \( a\) , \( -12 a - 188\) , \( -27 a - 908\bigr] \)	${y}^2+{x}{y}+a{y}={x}^3-a{x}^2+\left(-12a-188\right){x}-27a-908$

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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.