Elliptic curves over \(\Q(\sqrt{17}) \) of conductor 64.5

Base field	Conductor norm	Rank*	Torsion order	CM discriminant
Base field degree/signature	Bad \(p\)	$\Q$-curves	Torsion structure	CM
Analytic order of Ш*	Cyclic isogeny degree	Isogeny class size	Reduction	Galois image
j-invariant	Regulator*	Curves per isogeny class	Isogeny class degree	Nonmax$\ \ell$

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Results (8 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
64.5-a1	64.5-a	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{13} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$7.156385278$	0.867839188	\( -\frac{217}{16} a - \frac{339}{16} \)	\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( a + 2\) , \( 3 a - 4\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(a+2\right){x}+3a-4$
64.5-a2	64.5-a	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{13} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$7.156385278$	0.867839188	\( -\frac{1592342311}{2} a + \frac{4078872403}{2} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -16 a - 25\) , \( -4 a - 9\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16a-25\right){x}-4a-9$
64.5-a3	64.5-a	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{8} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{2} \)	$1$	$14.31277055$	0.867839188	\( \frac{159495}{4} a + \frac{481229}{4} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -11 a - 15\) , \( -33 a - 51\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11a-15\right){x}-33a-51$
64.5-a4	64.5-a	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{10} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$7.156385278$	0.867839188	\( \frac{23841914775}{2} a + \frac{37230413581}{2} \)	\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -171 a - 265\) , \( -1835 a - 2865\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-171a-265\right){x}-1835a-2865$
64.5-b1	64.5-b	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{13} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$14.01972920$	1.700141892	\( -\frac{217}{16} a - \frac{339}{16} \)	\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -2 a - 3\) , \( 2 a + 3\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2a-3\right){x}+2a+3$
64.5-b2	64.5-b	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{13} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$7.009864600$	1.700141892	\( -\frac{1592342311}{2} a + \frac{4078872403}{2} \)	\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 38 a - 95\) , \( 180 a - 461\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(38a-95\right){x}+180a-461$
64.5-b3	64.5-b	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{8} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{2} \)	$1$	$28.03945840$	1.700141892	\( \frac{159495}{4} a + \frac{481229}{4} \)	\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 3 a - 5\) , \( 3 a - 7\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(3a-5\right){x}+3a-7$
64.5-b4	64.5-b	$4$	$4$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	64.5	\( 2^{6} \)	\( 2^{10} \)	$1.04210$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 1 \)	$1$	$28.03945840$	1.700141892	\( \frac{23841914775}{2} a + \frac{37230413581}{2} \)	\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 3 a - 15\) , \( -3 a + 11\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(3a-15\right){x}-3a+11$

 

  *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.