Elliptic curves over \(\Q(\sqrt{17}) \) of conductor 128.2

Results (18 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
128.2-a1	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{22} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$4$	\( 2 \)	$1$	$2.710189503$	1.314635010	\( -292271882500 a + 748669862250 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 81 a - 220\) , \( 596 a - 1488\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(81a-220\right){x}+596a-1488$
128.2-a2	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{14} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$21.68151603$	1.314635010	\( -20500 a + 54000 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 16 a + 25\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a+25\right){x}$
128.2-a3	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{16} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2^{2} \)	$1$	$21.68151603$	1.314635010	\( 2000 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a\) , \( 0\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+a{x}$
128.2-a4	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{14} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$1$	$10.84075801$	1.314635010	\( 20500 a + 33500 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -14 a + 40\) , \( 15 a - 40\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-14a+40\right){x}+15a-40$
128.2-a5	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{20} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2^{3} \)	$1$	$10.84075801$	1.314635010	\( 1098500 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a - 20\) , \( 12 a - 16\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a-20\right){x}+12a-16$
128.2-a6	128.2-a	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{22} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$5.420379007$	1.314635010	\( 292271882500 a + 456397979750 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -79 a - 140\) , \( 516 a + 752\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-79a-140\right){x}+516a+752$
128.2-b1	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{16} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$23.36225846$	1.416544989	\( -7659605 a + 19620476 \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 28 a - 71\) , \( -142 a + 364\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(28a-71\right){x}-142a+364$
128.2-b2	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{23} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$5.840564617$	1.416544989	\( 343 a + 686 \)	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 37 a + 60\) , \( -38 a - 60\bigr] \)	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(37a+60\right){x}-38a-60$
128.2-b3	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{22} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2Cs	$1$	\( 2^{3} \)	$1$	$11.68112923$	1.416544989	\( -1995 a + 7016 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a - 7\) , \( 2 a - 6\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(3a-7\right){x}+2a-6$
128.2-b4	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{20} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2Cs	$1$	\( 2^{4} \)	$1$	$23.36225846$	1.416544989	\( 24225 a + 44228 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -33 a - 51\) , \( 178 a + 278\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(-33a-51\right){x}+178a+278$
128.2-b5	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{23} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$4$	\( 2 \)	$1$	$2.920282308$	1.416544989	\( -21069823 a + 54821634 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 43 a - 127\) , \( 218 a - 606\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(43a-127\right){x}+218a-606$
128.2-b6	128.2-b	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{22} \)	$1.23927$	$(-a+2), (-a-1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$				✓	$2$	2B	$1$	\( 2^{3} \)	$1$	$11.68112923$	1.416544989	\( 2701312025 a + 4218241728 \)	\( \bigl[0\) , \( -a\) , \( 0\) , \( -533 a - 831\) , \( 10078 a + 15738\bigr] \)	${y}^2={x}^{3}-a{x}^{2}+\left(-533a-831\right){x}+10078a+15738$
128.2-c1	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{15} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 1 \)	$2.324603913$	$6.204671930$	1.749094730	\( -774198 a + 1983150 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 9 a - 23\) , \( 21 a - 54\bigr] \)	${y}^2={x}^{3}+\left(9a-23\right){x}+21a-54$
128.2-c2	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{21} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{2} \)	$1.162301956$	$12.40934386$	1.749094730	\( -349618194 a + 895566564 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 192 a - 491\) , \( -2076 a + 5318\bigr] \)	${y}^2={x}^{3}+\left(192a-491\right){x}-2076a+5318$
128.2-c3	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{18} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2^{4} \)	$0.581150978$	$24.81868772$	1.749094730	\( -8748 a + 25056 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( 12 a - 31\) , \( -32 a + 82\bigr] \)	${y}^2={x}^{3}+\left(12a-31\right){x}-32a+82$
128.2-c4	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( 2^{18} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2Cs	$1$	\( 2^{2} \)	$1.162301956$	$12.40934386$	1.749094730	\( 8748 a + 16308 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -12 a - 19\) , \( -32 a - 50\bigr] \)	${y}^2={x}^{3}+\left(-12a-19\right){x}-32a-50$
128.2-c5	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{15} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2^{3} \)	$0.290575489$	$24.81868772$	1.749094730	\( 774198 a + 1208952 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -9 a - 14\) , \( 21 a + 33\bigr] \)	${y}^2={x}^{3}+\left(-9a-14\right){x}+21a+33$
128.2-c6	128.2-c	$6$	$8$	\(\Q(\sqrt{17}) \)	$2$	$[2, 0]$	128.2	\( 2^{7} \)	\( - 2^{21} \)	$1.23927$	$(-a+2), (-a-1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$2.324603913$	$3.102335965$	1.749094730	\( 349618194 a + 545948370 \)	\( \bigl[0\) , \( 0\) , \( 0\) , \( -192 a - 299\) , \( -2076 a - 3242\bigr] \)	${y}^2={x}^{3}+\left(-192a-299\right){x}-2076a-3242$

 

  *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.