Elliptic curves over 3.3.316.1 of conductor 128.7

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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
128.7-a1	128.7-a	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{25} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \cdot 3 \cdot 5 \)	$0.109884051$	$26.65027186$	3.706596283	\( \frac{10610687801034935261}{256} a^{2} - \frac{29854300194558708471}{256} a + \frac{1462650220091725987}{32} \)	\( \bigl[a^{2} - 3\) , \( -a - 1\) , \( a^{2} - 3\) , \( -147 a^{2} + 585 a - 567\) , \( 3709 a^{2} - 10808 a + 4961\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-147a^{2}+585a-567\right){x}+3709a^{2}-10808a+4961$
128.7-a2	128.7-a	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{15} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \cdot 5 \)	$0.036628017$	$239.8524467$	3.706596283	\( -\frac{1780690961}{8} a^{2} + \frac{941301753}{8} a + \frac{3785006083}{4} \)	\( \bigl[a^{2} - 3\) , \( -a - 1\) , \( a^{2} - 3\) , \( 58 a^{2} - 25 a - 262\) , \( -357 a^{2} + 178 a + 1541\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(58a^{2}-25a-262\right){x}-357a^{2}+178a+1541$
128.7-a3	128.7-a	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{44} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \cdot 3 \cdot 5 \)	$0.054942025$	$13.32513593$	3.706596283	\( -\frac{541255732825}{8192} a^{2} + \frac{6091495909813}{32768} a - \frac{2387470670643}{32768} \)	\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 3\) , \( -873 a^{2} + 2452 a - 952\) , \( 30154 a^{2} - 84843 a + 33257\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-873a^{2}+2452a-952\right){x}+30154a^{2}-84843a+33257$
128.7-a4	128.7-a	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{24} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{3} \cdot 5 \)	$0.018314008$	$119.9262233$	3.706596283	\( \frac{56079}{8} a^{2} - \frac{116333}{32} a - \frac{958845}{32} \)	\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 3\) , \( -8 a^{2} + 22 a - 7\) , \( 63 a^{2} - 176 a + 67\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-8a^{2}+22a-7\right){x}+63a^{2}-176a+67$
128.7-b1	128.7-b	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{20} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$19.11536543$	1.075323318	\( -\frac{3655}{4} a^{2} - \frac{775}{2} a + \frac{7961}{4} \)	\( \bigl[a^{2} - 3\) , \( a^{2} - a - 3\) , \( 0\) , \( -4 a^{2} + 2 a + 18\) , \( 0\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-4a^{2}+2a+18\right){x}$
128.7-b2	128.7-b	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{11} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$38.23073087$	1.075323318	\( \frac{112062878879}{2} a^{2} - \frac{315301616071}{2} a + 61790982411 \)	\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 4\) , \( a^{2} - 3\) , \( 34420 a^{2} - 18220 a - 146255\) , \( -5017146 a^{2} + 2655658 a + 21318558\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(34420a^{2}-18220a-146255\right){x}-5017146a^{2}+2655658a+21318558$
128.7-b3	128.7-b	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{17} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2 \)	$1$	$9.557682717$	1.075323318	\( \frac{118715864416425}{2} a^{2} + \frac{159426274721303}{2} a - 50669979335827 \)	\( \bigl[a + 1\) , \( a^{2} - a - 2\) , \( a + 1\) , \( 2032 a^{2} - 1081 a - 8641\) , \( 71339 a^{2} - 37775 a - 303154\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2032a^{2}-1081a-8641\right){x}+71339a^{2}-37775a-303154$
128.7-b4	128.7-b	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{16} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{3} \)	$1$	$38.23073087$	1.075323318	\( 2838366 a^{2} + \frac{6985779}{2} a - \frac{4542013}{2} \)	\( \bigl[a + 1\) , \( a^{2} - a - 2\) , \( a + 1\) , \( 217 a^{2} - 116 a - 921\) , \( -797 a^{2} + 421 a + 3386\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(217a^{2}-116a-921\right){x}-797a^{2}+421a+3386$
128.7-c1	128.7-c	$2$	$2$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{8} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$93.82891037$	2.639144295	\( -1346 a^{2} + \frac{7699}{2} a - \frac{3261}{2} \)	\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 3\) , \( a + 1\) , \( -2 a^{2} + a + 8\) , \( 10 a^{2} + 15 a - 7\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-2a^{2}+a+8\right){x}+10a^{2}+15a-7$
128.7-c2	128.7-c	$2$	$2$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{13} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$93.82891037$	2.639144295	\( \frac{39790591}{2} a^{2} - \frac{112259079}{2} a + 22350587 \)	\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 3\) , \( a + 1\) , \( -42 a^{2} - 44 a + 38\) , \( 282 a^{2} + 401 a - 249\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-42a^{2}-44a+38\right){x}+282a^{2}+401a-249$
128.7-d1	128.7-d	$2$	$2$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{8} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$81.04344650$	2.279525027	\( -1346 a^{2} + \frac{7699}{2} a - \frac{3261}{2} \)	\( \bigl[a + 1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( a + 1\) , \( -11 a^{2} - 15 a + 9\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a+1\right){x}-11a^{2}-15a+9$
128.7-d2	128.7-d	$2$	$2$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{13} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$1$	$81.04344650$	2.279525027	\( \frac{39790591}{2} a^{2} - \frac{112259079}{2} a + 22350587 \)	\( \bigl[a + 1\) , \( a^{2} - 2\) , \( a^{2} + a - 2\) , \( -40 a^{2} - 44 a + 31\) , \( -323 a^{2} - 446 a + 281\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-40a^{2}-44a+31\right){x}-323a^{2}-446a+281$
128.7-e1	128.7-e	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{17} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$0.229099761$	$102.7552442$	1.986444107	\( \frac{118715864416425}{2} a^{2} + \frac{159426274721303}{2} a - 50669979335827 \)	\( \bigl[a^{2} - 3\) , \( -a^{2} + 3\) , \( 0\) , \( 2030 a^{2} - 1079 a - 8634\) , \( -69308 a^{2} + 36695 a + 294516\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(2030a^{2}-1079a-8634\right){x}-69308a^{2}+36695a+294516$
128.7-e2	128.7-e	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{16} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{3} \)	$0.114549880$	$205.5104884$	1.986444107	\( 2838366 a^{2} + \frac{6985779}{2} a - \frac{4542013}{2} \)	\( \bigl[a^{2} - 3\) , \( -a^{2} + 3\) , \( 0\) , \( 215 a^{2} - 114 a - 914\) , \( 1013 a^{2} - 536 a - 4304\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(215a^{2}-114a-914\right){x}+1013a^{2}-536a-4304$
128.7-e3	128.7-e	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{11} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2 \)	$0.229099761$	$102.7552442$	1.986444107	\( \frac{112062878879}{2} a^{2} - \frac{315301616071}{2} a + 61790982411 \)	\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 34422 a^{2} - 18218 a - 146260\) , \( 5051567 a^{2} - 2673877 a - 21464816\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(34422a^{2}-18218a-146260\right){x}+5051567a^{2}-2673877a-21464816$
128.7-e4	128.7-e	$4$	$4$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{20} \)	$3.56602$	$(a), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$0.057274940$	$102.7552442$	1.986444107	\( -\frac{3655}{4} a^{2} - \frac{775}{2} a + \frac{7961}{4} \)	\( \bigl[a + 1\) , \( -a^{2} + 4\) , \( a + 1\) , \( -6 a^{2} + 2 a + 23\) , \( -5 a^{2} + 2 a + 20\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-6a^{2}+2a+23\right){x}-5a^{2}+2a+20$
128.7-f1	128.7-f	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{44} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 3 \cdot 5 \)	$1$	$3.130107050$	2.641234178	\( -\frac{541255732825}{8192} a^{2} + \frac{6091495909813}{32768} a - \frac{2387470670643}{32768} \)	\( \bigl[a^{2} - 3\) , \( -a\) , \( a^{2} + a - 2\) , \( -873 a^{2} + 2450 a - 955\) , \( -31027 a^{2} + 87293 a - 34213\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-a{x}^{2}+\left(-873a^{2}+2450a-955\right){x}-31027a^{2}+87293a-34213$
128.7-f2	128.7-f	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( - 2^{24} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2^{2} \cdot 5 \)	$1$	$9.390321152$	2.641234178	\( \frac{56079}{8} a^{2} - \frac{116333}{32} a - \frac{958845}{32} \)	\( \bigl[a^{2} - 3\) , \( -a\) , \( a^{2} + a - 2\) , \( -8 a^{2} + 20 a - 10\) , \( -71 a^{2} + 196 a - 78\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-a{x}^{2}+\left(-8a^{2}+20a-10\right){x}-71a^{2}+196a-78$
128.7-f3	128.7-f	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{25} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \cdot 3 \cdot 5 \)	$1$	$6.260214101$	2.641234178	\( \frac{10610687801034935261}{256} a^{2} - \frac{29854300194558708471}{256} a + \frac{1462650220091725987}{32} \)	\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -148 a^{2} + 585 a - 564\) , \( -3709 a^{2} + 10807 a - 4963\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-148a^{2}+585a-564\right){x}-3709a^{2}+10807a-4963$
128.7-f4	128.7-f	$4$	$6$	3.3.316.1	$3$	$[3, 0]$	128.7	\( 2^{7} \)	\( 2^{15} \)	$3.56602$	$(a), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	\( 2 \cdot 5 \)	$1$	$18.78064230$	2.641234178	\( -\frac{1780690961}{8} a^{2} + \frac{941301753}{8} a + \frac{3785006083}{4} \)	\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 57 a^{2} - 25 a - 259\) , \( 357 a^{2} - 179 a - 1543\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(57a^{2}-25a-259\right){x}+357a^{2}-179a-1543$

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  *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.