Elliptic curves over \(\Q(\sqrt{17}) \) of conductor 32.4

Results (12 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
32.4-a1	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{39} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 3 \)	$1$	$1.942891550$	1.413661249	\( -\frac{55573026649}{16777216} a - \frac{85738862931}{16777216} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -23 a + 52\) , \( -405 a + 1032\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-23a+52\right){x}-405a+1032$
32.4-a2	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{21} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{5} \)	$1$	$11.65734930$	1.413661249	\( -\frac{110887}{256} a + \frac{66933}{64} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -7 a - 12\) , \( 67 a + 104\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-7a-12\right){x}+67a+104$
32.4-a3	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{39} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{5} \cdot 3 \)	$1$	$3.885783100$	1.413661249	\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 58 a + 88\) , \( -1783 a - 2784\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(58a+88\right){x}-1783a-2784$
32.4-a4	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{21} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$1$	$5.828674651$	1.413661249	\( \frac{110887}{256} a + \frac{156845}{256} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 2 a - 8\) , \( 15 a - 40\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a-8\right){x}+15a-40$
32.4-a5	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{18} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	\( 2^{3} \)	$1$	$11.65734930$	1.413661249	\( -\frac{915957}{16} a + \frac{2374013}{16} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -7 a - 12\) , \( -7 a - 12\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-7a-12\right){x}-7a-12$
32.4-a6	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{21} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 3 \)	$1$	$7.771566201$	1.413661249	\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \)	\( \bigl[a\) , \( a\) , \( a\) , \( 2388 a - 6147\) , \( -92365 a + 236610\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(2388a-6147\right){x}-92365a+236610$
32.4-a7	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{30} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	\( 2^{4} \cdot 3 \)	$1$	$7.771566201$	1.413661249	\( -\frac{203862548967}{4096} a + \frac{130566616997}{1024} \)	\( \bigl[a\) , \( a\) , \( a\) , \( 148 a - 387\) , \( -1485 a + 3778\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(148a-387\right){x}-1485a+3778$
32.4-a8	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{18} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	\( 2^{4} \)	$1$	$23.31469860$	1.413661249	\( \frac{915957}{16} a + \frac{182257}{2} \)	\( \bigl[a\) , \( a\) , \( a\) , \( 3 a - 7\) , \( -a + 2\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(3a-7\right){x}-a+2$
32.4-a9	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{15} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	\( 2 \)	$1$	$2.914337325$	1.413661249	\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -77 a - 132\) , \( -619 a - 988\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-77a-132\right){x}-619a-988$
32.4-a10	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{30} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2Cs, 3B	$1$	\( 2^{3} \cdot 3 \)	$1$	$3.885783100$	1.413661249	\( \frac{203862548967}{4096} a + \frac{318403919021}{4096} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -402 a - 632\) , \( -6673 a - 10420\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-402a-632\right){x}-6673a-10420$
32.4-a11	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{15} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$1$	\( 2^{2} \)	$1$	$23.31469860$	1.413661249	\( \frac{54503407609}{4} a + \frac{42555672073}{2} \)	\( \bigl[a\) , \( a\) , \( a\) , \( 23 a - 87\) , \( -129 a + 354\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(23a-87\right){x}-129a+354$
32.4-a12	32.4-a	$12$	$24$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	32.4	\( 2^{5} \)	\( 2^{21} \)	$0.87630$	$(-a+2), (-a-1)$	$0$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 3$	2B, 3B	$4$	\( 2 \cdot 3 \)	$1$	$0.971445775$	1.413661249	\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -6442 a - 10072\) , \( -410033 a - 640308\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-6442a-10072\right){x}-410033a-640308$

 

  *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.