Elliptic curves over 3.3.169.1 of conductor 125.6

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 2059 over totally real cubic fields with discriminant 1957

Results (12 matches)

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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
125.6-a1	125.6-a	$4$	$4$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{11} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$1$	$5.187542918$	1.596167051	\( \frac{53152820281148}{625} a^{2} + \frac{88975367883001}{625} a + \frac{21722839234996}{625} \)	\( \bigl[a^{2} - a - 3\) , \( a - 1\) , \( a + 1\) , \( -320 a^{2} - 1030 a - 810\) , \( -13157 a^{2} - 26340 a - 11321\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-320a^{2}-1030a-810\right){x}-13157a^{2}-26340a-11321$
125.6-a2	125.6-a	$4$	$4$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{10} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{3} \)	$1$	$41.50034334$	1.596167051	\( \frac{217317341}{5} a^{2} - \frac{2574843704}{25} a - \frac{785560232}{25} \)	\( \bigl[a^{2} - a - 3\) , \( a - 1\) , \( a + 1\) , \( -20 a^{2} - 65 a - 50\) , \( -221 a^{2} - 440 a - 187\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-20a^{2}-65a-50\right){x}-221a^{2}-440a-187$
125.6-a3	125.6-a	$4$	$4$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( - 5^{11} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$20.75017167$	1.596167051	\( \frac{5230339383024320708}{625} a^{2} - \frac{2486715541714430821}{125} a - \frac{3797798826756471012}{625} \)	\( \bigl[a^{2} - a - 3\) , \( a - 1\) , \( a + 1\) , \( -40 a^{2} - 60 a - 10\) , \( -61 a^{2} - 320 a - 317\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-40a^{2}-60a-10\right){x}-61a^{2}-320a-317$
125.6-a4	125.6-a	$4$	$4$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{8} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$83.00068669$	1.596167051	\( -\frac{18759}{5} a^{2} + \frac{39689}{5} a + \frac{24453}{5} \)	\( \bigl[a^{2} - a - 3\) , \( a - 1\) , \( a + 1\) , \( -5 a - 5\) , \( -7 a^{2} - 10 a - 1\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-5\right){x}-7a^{2}-10a-1$
125.6-b1	125.6-b	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{19} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.1	$1$	\( 2^{2} \)	$1$	$11.00012459$	0.846163430	\( -\frac{21225835996837}{9765625} a^{2} + \frac{27017382271681}{9765625} a + \frac{77573965543901}{9765625} \)	\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 85 a^{2} - 113 a - 320\) , \( 682 a^{2} - 873 a - 2497\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(85a^{2}-113a-320\right){x}+682a^{2}-873a-2497$
125.6-b2	125.6-b	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{14} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.1	$1$	\( 2 \)	$1$	$22.00024918$	0.846163430	\( -\frac{2843862}{3125} a^{2} + \frac{4771131}{3125} a + \frac{12367976}{3125} \)	\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 5 a^{2} - 8 a - 20\) , \( 14 a^{2} - 19 a - 53\bigr] \)	${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(5a^{2}-8a-20\right){x}+14a^{2}-19a-53$
125.6-b3	125.6-b	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{11} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.2	$1$	\( 2^{2} \)	$1$	$11.00012459$	0.846163430	\( \frac{917921432246848}{25} a^{2} + \frac{1515529783362901}{25} a + \frac{346230473004096}{25} \)	\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 2\) , \( 453 a^{2} - 785 a - 1954\) , \( -9087 a^{2} + 8906 a + 29516\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}+\left(453a^{2}-785a-1954\right){x}-9087a^{2}+8906a+29516$
125.6-b4	125.6-b	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{10} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.4.2	$1$	\( 2 \)	$1$	$22.00024918$	0.846163430	\( -\frac{116413326867}{5} a^{2} + \frac{148272075696}{5} a + \frac{425000233856}{5} \)	\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 2\) , \( 523 a^{2} - 680 a - 1929\) , \( -8214 a^{2} + 10407 a + 29912\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}+\left(523a^{2}-680a-1929\right){x}-8214a^{2}+10407a+29912$
125.6-c1	125.6-c	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{13} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	$1$	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.1.1	$1$	\( 2^{2} \cdot 5 \)	$0.376082831$	$90.63000163$	1.573125580	\( -\frac{21225835996837}{9765625} a^{2} + \frac{27017382271681}{9765625} a + \frac{77573965543901}{9765625} \)	\( \bigl[a^{2} - a - 3\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 3\) , \( 55 a^{2} - 72 a - 203\) , \( -288 a^{2} + 366 a + 1050\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(55a^{2}-72a-203\right){x}-288a^{2}+366a+1050$
125.6-c2	125.6-c	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{8} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	$1$	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.1.1	$1$	\( 2 \cdot 5 \)	$0.752165662$	$90.63000163$	1.573125580	\( -\frac{2843862}{3125} a^{2} + \frac{4771131}{3125} a + \frac{12367976}{3125} \)	\( \bigl[a^{2} - a - 3\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 3\) , \( 5 a^{2} - 7 a - 18\) , \( -4 a^{2} + 4 a + 13\bigr] \)	${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(5a^{2}-7a-18\right){x}-4a^{2}+4a+13$
125.6-c3	125.6-c	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{5} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.1.2	$1$	\( 2^{2} \)	$1.880414155$	$3.625200065$	1.573125580	\( \frac{917921432246848}{25} a^{2} + \frac{1515529783362901}{25} a + \frac{346230473004096}{25} \)	\( \bigl[a^{2} - a - 2\) , \( a^{2} - 4\) , \( 1\) , \( 298 a^{2} - 455 a - 1194\) , \( 3483 a^{2} - 5075 a - 13596\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(298a^{2}-455a-1194\right){x}+3483a^{2}-5075a-13596$
125.6-c4	125.6-c	$4$	$10$	3.3.169.1	$3$	$[3, 0]$	125.6	\( 5^{3} \)	\( 5^{4} \)	$2.59757$	$(-a^2+2a+3), (-a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 5$	2B, 5B.1.2	$1$	\( 2 \)	$3.760828310$	$3.625200065$	1.573125580	\( -\frac{116413326867}{5} a^{2} + \frac{148272075696}{5} a + \frac{425000233856}{5} \)	\( \bigl[a^{2} - a - 2\) , \( a^{2} - 4\) , \( 1\) , \( 323 a^{2} - 415 a - 1184\) , \( 3704 a^{2} - 4733 a - 13544\bigr] \)	${y}^2+\left(a^{2}-a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(323a^{2}-415a-1184\right){x}+3704a^{2}-4733a-13544$

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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.