Elliptic curves over 6.6.1279733.1 of conductor 29.3

Results (6 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
29.3-a1	29.3-a	$1$	$1$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29^{3} \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3Ns	$1$	\( 3 \)	$0.002830732$	$70594.46889$	3.17967	\( -\frac{86278465922}{24389} a^{5} - \frac{47189236829}{24389} a^{4} + \frac{438778368281}{24389} a^{3} + \frac{277275142138}{24389} a^{2} - \frac{358957259814}{24389} a - \frac{88742923489}{24389} \)	\( \bigl[a^{5} - 2 a^{4} - 2 a^{3} + 5 a^{2} - 2 a - 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{4} - 5 a^{2} + 4\) , \( a^{5} + 4 a^{4} - 7 a^{3} - 16 a^{2} + 7 a + 4\) , \( 4 a^{5} + 2 a^{4} - 24 a^{3} - 9 a^{2} + 29 a - 1\bigr] \)	${y}^2+\left(a^{5}-2a^{4}-2a^{3}+5a^{2}-2a-1\right){x}{y}+\left(a^{4}-5a^{2}+4\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(a^{5}+4a^{4}-7a^{3}-16a^{2}+7a+4\right){x}+4a^{5}+2a^{4}-24a^{3}-9a^{2}+29a-1$
29.3-b1	29.3-b	$2$	$5$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29^{5} \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$5$	5B.4.2	$1$	\( 5 \)	$0.072380082$	$1279.479688$	2.45592	\( -\frac{34286919698837807988428}{20511149} a^{5} - \frac{11384698874034008601269}{20511149} a^{4} + \frac{179179369561896340242986}{20511149} a^{3} + \frac{74989014479850097421206}{20511149} a^{2} - \frac{168020705700556465895183}{20511149} a - \frac{14701659055200981175163}{20511149} \)	\( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{5} - 2 a^{4} - 2 a^{3} + 4 a^{2} - a + 2\) , \( a^{2} - a - 2\) , \( 78 a^{5} - 344 a^{4} + 69 a^{3} + 933 a^{2} - 534 a - 149\) , \( 889 a^{5} - 3638 a^{4} + 2347 a^{3} + 7042 a^{2} - 10634 a + 4016\bigr] \)	${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{5}-2a^{4}-2a^{3}+4a^{2}-a+2\right){x}^{2}+\left(78a^{5}-344a^{4}+69a^{3}+933a^{2}-534a-149\right){x}+889a^{5}-3638a^{4}+2347a^{3}+7042a^{2}-10634a+4016$
29.3-b2	29.3-b	$2$	$5$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29 \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$5$	5B.4.1	$1$	\( 1 \)	$0.014476016$	$31986.99220$	2.45592	\( \frac{23970}{29} a^{5} - \frac{73281}{29} a^{4} - \frac{77587}{29} a^{3} + \frac{268660}{29} a^{2} + \frac{49483}{29} a - \frac{112985}{29} \)	\( \bigl[a^{5} - 2 a^{4} - 2 a^{3} + 6 a^{2} - 4 a - 3\) , \( -a^{5} + a^{4} + 5 a^{3} - 2 a^{2} - 5 a - 2\) , \( 0\) , \( -2 a^{5} + 2 a^{4} + 15 a^{3} - 6 a^{2} - 29 a - 3\) , \( 2 a^{5} - 2 a^{4} - 14 a^{3} + 7 a^{2} + 27 a + 2\bigr] \)	${y}^2+\left(a^{5}-2a^{4}-2a^{3}+6a^{2}-4a-3\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-2a^{2}-5a-2\right){x}^{2}+\left(-2a^{5}+2a^{4}+15a^{3}-6a^{2}-29a-3\right){x}+2a^{5}-2a^{4}-14a^{3}+7a^{2}+27a+2$
29.3-c1	29.3-c	$2$	$5$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29^{5} \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$5$	5B.1.2	$625$	\( 1 \)	$1$	$3.987160790$	2.20285	\( -\frac{34286919698837807988428}{20511149} a^{5} - \frac{11384698874034008601269}{20511149} a^{4} + \frac{179179369561896340242986}{20511149} a^{3} + \frac{74989014479850097421206}{20511149} a^{2} - \frac{168020705700556465895183}{20511149} a - \frac{14701659055200981175163}{20511149} \)	\( \bigl[a^{2} - 2\) , \( a^{5} - 2 a^{4} - 4 a^{3} + 6 a^{2} + 5 a - 2\) , \( a^{5} - 2 a^{4} - 2 a^{3} + 6 a^{2} - 3 a - 2\) , \( 65 a^{5} - 151 a^{4} - 347 a^{3} + 706 a^{2} + 535 a - 744\) , \( 8693 a^{5} - 18199 a^{4} - 50436 a^{3} + 91318 a^{2} + 78787 a - 102152\bigr] \)	${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{5}-2a^{4}-2a^{3}+6a^{2}-3a-2\right){y}={x}^{3}+\left(a^{5}-2a^{4}-4a^{3}+6a^{2}+5a-2\right){x}^{2}+\left(65a^{5}-151a^{4}-347a^{3}+706a^{2}+535a-744\right){x}+8693a^{5}-18199a^{4}-50436a^{3}+91318a^{2}+78787a-102152$
29.3-c2	29.3-c	$2$	$5$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29 \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	0	$\Z/5\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$5$	5B.1.1	$1$	\( 1 \)	$1$	$62299.38735$	2.20285	\( \frac{23970}{29} a^{5} - \frac{73281}{29} a^{4} - \frac{77587}{29} a^{3} + \frac{268660}{29} a^{2} + \frac{49483}{29} a - \frac{112985}{29} \)	\( \bigl[a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 6 a\) , \( a^{3} - 3 a\) , \( 2 a^{5} - a^{4} - 11 a^{3} + 2 a^{2} + 14 a + 1\) , \( 0\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+6a\right){x}^{2}+\left(2a^{5}-a^{4}-11a^{3}+2a^{2}+14a+1\right){x}$
29.3-d1	29.3-d	$1$	$1$	6.6.1279733.1	$6$	$[6, 0]$	29.3	\( 29 \)	\( 29^{3} \)	$133.83347$	$(a^4-a^3-3a^2+a-2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3Ns	$1$	\( 1 \)	$1$	$1581.476661$	1.39799	\( -\frac{86278465922}{24389} a^{5} - \frac{47189236829}{24389} a^{4} + \frac{438778368281}{24389} a^{3} + \frac{277275142138}{24389} a^{2} - \frac{358957259814}{24389} a - \frac{88742923489}{24389} \)	\( \bigl[a^{5} - a^{4} - 4 a^{3} + 2 a^{2} + 2 a\) , \( -a^{5} + a^{4} + 3 a^{3} - 2 a^{2} + 3 a\) , \( a^{5} - a^{4} - 3 a^{3} + a^{2} - a + 2\) , \( -a^{5} - 2 a^{4} + 5 a^{3} + 12 a^{2} - 3 a - 7\) , \( -a^{4} - a^{3} + 5 a^{2} + 5 a - 7\bigr] \)	${y}^2+\left(a^{5}-a^{4}-4a^{3}+2a^{2}+2a\right){x}{y}+\left(a^{5}-a^{4}-3a^{3}+a^{2}-a+2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+3a^{3}-2a^{2}+3a\right){x}^{2}+\left(-a^{5}-2a^{4}+5a^{3}+12a^{2}-3a-7\right){x}-a^{4}-a^{3}+5a^{2}+5a-7$

 

  *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.