Elliptic curves over \(\Q(\sqrt{489}) \)

Results (4 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
4.2-a1	4.2-a	$2$	$3$	\(\Q(\sqrt{489}) \)	$2$	$[2, 0]$	4.2	\( 2^{2} \)	\( 2^{8} \)	$2.79452$	$(9a+95)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$163$	163Ns.4.1	$1$	\( 1 \)	$1.888823678$	$17.69503190$	3.022862093	\( 0 \)	\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 41\) , \( -6386120019080735662903 a - 67416175631366988477355\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+41\right){x}-6386120019080735662903a-67416175631366988477355$
4.2-a2	4.2-a	$2$	$3$	\(\Q(\sqrt{489}) \)	$2$	$[2, 0]$	4.2	\( 2^{2} \)	\( 2^{8} \)	$2.79452$	$(9a+95)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$163$	163Ns.4.1	$1$	\( 3 \)	$0.629607892$	$17.69503190$	3.022862093	\( 0 \)	\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 41\) , \( 41524 a - 479992\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+41\right){x}+41524a-479992$
4.3-a1	4.3-a	$2$	$3$	\(\Q(\sqrt{489}) \)	$2$	$[2, 0]$	4.3	\( 2^{2} \)	\( 2^{8} \)	$2.79452$	$(9a-104)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$163$	163Ns.4.1	$1$	\( 1 \)	$1.888823678$	$17.69503190$	3.022862093	\( 0 \)	\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 41\) , \( 6386120019080735662902 a - 73802295650447724140298\bigr] \)	${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+41\right){x}+6386120019080735662902a-73802295650447724140298$
4.3-a2	4.3-a	$2$	$3$	\(\Q(\sqrt{489}) \)	$2$	$[2, 0]$	4.3	\( 2^{2} \)	\( 2^{8} \)	$2.79452$	$(9a-104)$	$1$	$\mathsf{trivial}$	$\textsf{potential}$	$-3$	$N(\mathrm{U}(1))$	✓			✓	$163$	163Ns.4.1	$1$	\( 3 \)	$0.629607892$	$17.69503190$	3.022862093	\( 0 \)	\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 41\) , \( -41525 a - 438426\bigr] \)	${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+41\right){x}-41525a-438426$

 

  *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.