Elliptic curves over \(\Q(\sqrt{-2}) \) of conductor 20000.1

Results (20 matches)

 

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
20000.1-a1	20000.1-a	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$0.813415655$	$0.820904889$	3.777290258	\( -\frac{64}{25} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -8\) , \( 238\bigr] \)	${y}^2={x}^{3}+{x}^{2}-8{x}+238$
20000.1-a2	20000.1-a	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{14} \)	$3.00567$	$(a), (5)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$0.813415655$	$0.820904889$	3.777290258	\( \frac{438976}{5} \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -158\) , \( -812\bigr] \)	${y}^2={x}^{3}+{x}^{2}-158{x}-812$
20000.1-b1	20000.1-b	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2, 5$	2Cn, 5Ns	$1$	\( 2 \cdot 3 \)	$0.196377791$	$1.063939980$	1.772865336	\( -5000 \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -51\) , \( -176\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-51{x}-176$
20000.1-c1	20000.1-c	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{4} \)	$3.00567$	$(a), (5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2, 5$	2Cn, 5Ns	$1$	\( 2 \)	$0.218002349$	$5.319699904$	3.280146962	\( -5000 \)	\( \bigl[a\) , \( 1\) , \( 0\) , \( -2\) , \( 1\bigr] \)	${y}^2+a{x}{y}={x}^{3}+{x}^{2}-2{x}+1$
20000.1-d1	20000.1-d	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{4} \)	$3.00567$	$(a), (5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cn	$1$	\( 2^{2} \)	$0.176456074$	$4.043909875$	4.036575423	\( -320 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( -a\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+{x}-a$
20000.1-e1	20000.1-e	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cn	$1$	\( 2 \)	$1$	$0.808781975$	1.143790438	\( -320 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 41\) , \( 199 a\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+41{x}+199a$
20000.1-f1	20000.1-f	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{6} \)	$3.00567$	$(a), (5)$	$2$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$0.713078665$	$3.074676569$	6.201287618	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 5\) , \( 0\bigr] \)	${y}^2={x}^{3}+5{x}$
20000.1-f2	20000.1-f	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{6} \)	$3.00567$	$(a), (5)$	$2$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$0.713078665$	$3.074676569$	6.201287618	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -5\) , \( 0\bigr] \)	${y}^2={x}^{3}-5{x}$
20000.1-g1	20000.1-g	$4$	$4$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{12} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{3} \)	$0.949741086$	$1.375037163$	3.693725825	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 25\) , \( 0\bigr] \)	${y}^2={x}^{3}+25{x}$
20000.1-g2	20000.1-g	$4$	$4$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{12} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	\( 2^{4} \)	$1.899482172$	$1.375037163$	3.693725825	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \)	${y}^2={x}^{3}-25{x}$
20000.1-g3	20000.1-g	$4$	$4$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{12} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{3} \)	$0.949741086$	$1.375037163$	3.693725825	\( 287496 \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -68\) , \( 253\bigr] \)	${y}^2+a{x}{y}={x}^{3}-{x}^{2}-68{x}+253$
20000.1-g4	20000.1-g	$4$	$4$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{12} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{3} \)	$0.949741086$	$1.375037163$	3.693725825	\( 287496 \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -67\) , \( -184\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-67{x}-184$
20000.1-h1	20000.1-h	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{18} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{2} \)	$4.982067391$	$0.614935313$	4.332654213	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 125\) , \( 0\bigr] \)	${y}^2={x}^{3}+125{x}$
20000.1-h2	20000.1-h	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{18} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	\( 2^{3} \)	$2.491033695$	$0.614935313$	4.332654213	\( 1728 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -125\) , \( 0\bigr] \)	${y}^2={x}^{3}-125{x}$
20000.1-i1	20000.1-i	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cn	$1$	\( 2 \)	$1$	$0.808781975$	1.143790438	\( -320 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( 41\) , \( -199 a\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+41{x}-199a$
20000.1-j1	20000.1-j	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{4} \)	$3.00567$	$(a), (5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cn	$1$	\( 2^{2} \)	$0.176456074$	$4.043909875$	4.036575423	\( -320 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( a\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+{x}+a$
20000.1-k1	20000.1-k	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{4} \)	$3.00567$	$(a), (5)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2, 5$	2Cn, 5Ns	$1$	\( 2 \)	$0.260556717$	$5.319699904$	7.840872579	\( -5000 \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( -2\) , \( -1\bigr] \)	${y}^2+a{x}{y}={x}^{3}-2{x}-1$
20000.1-l1	20000.1-l	$1$	$1$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{6} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2, 5$	2Cn, 5Ns	$1$	\( 2 \cdot 3 \)	$1$	$1.063939980$	4.513915051	\( -5000 \)	\( \bigl[a\) , \( 0\) , \( a\) , \( -51\) , \( 177\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-51{x}+177$
20000.1-m1	20000.1-m	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{16} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{3} \)	$3.640840235$	$0.820904889$	8.453556469	\( -\frac{64}{25} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -8\) , \( -238\bigr] \)	${y}^2={x}^{3}-{x}^{2}-8{x}-238$
20000.1-m2	20000.1-m	$2$	$2$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	20000.1	\( 2^{5} \cdot 5^{4} \)	\( 2^{12} \cdot 5^{14} \)	$3.00567$	$(a), (5)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	\( 2^{4} \)	$1.820420117$	$0.820904889$	8.453556469	\( \frac{438976}{5} \)	\( \bigl[0\) , \( -1\) , \( 0\) , \( -158\) , \( 812\bigr] \)	${y}^2={x}^{3}-{x}^{2}-158{x}+812$

 

  *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.