Elliptic curves over 4.4.6125.1 of conductor 71.1

Results (8 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
71.1-a1	71.1-a	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71 \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$0.880646448$	$82.10556773$	3.695567517	\( \frac{337705309521}{71} a^{3} + \frac{407692977510}{71} a^{2} - \frac{2139474325813}{71} a - \frac{1682987986020}{71} \)	\( \bigl[2 a^{3} + 3 a^{2} - 11 a - 11\) , \( 2 a^{3} + 3 a^{2} - 13 a - 12\) , \( 1\) , \( 9 a^{3} + 14 a^{2} - 49 a - 51\) , \( 16 a^{3} + 20 a^{2} - 90 a - 74\bigr] \)	${y}^2+\left(2a^{3}+3a^{2}-11a-11\right){x}{y}+{y}={x}^{3}+\left(2a^{3}+3a^{2}-13a-12\right){x}^{2}+\left(9a^{3}+14a^{2}-49a-51\right){x}+16a^{3}+20a^{2}-90a-74$
71.1-b1	71.1-b	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71^{3} \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 3 \)	$0.278940957$	$74.56956939$	3.189349568	\( \frac{40567729198}{357911} a^{3} - \frac{18573487223}{357911} a^{2} - \frac{289892232122}{357911} a + \frac{246977548593}{357911} \)	\( \bigl[1\) , \( 2 a^{3} + 3 a^{2} - 11 a - 12\) , \( a^{3} + 2 a^{2} - 5 a - 8\) , \( 38 a^{3} + 46 a^{2} - 240 a - 187\) , \( 125 a^{3} + 151 a^{2} - 793 a - 627\bigr] \)	${y}^2+{x}{y}+\left(a^{3}+2a^{2}-5a-8\right){y}={x}^{3}+\left(2a^{3}+3a^{2}-11a-12\right){x}^{2}+\left(38a^{3}+46a^{2}-240a-187\right){x}+125a^{3}+151a^{2}-793a-627$
71.1-c1	71.1-c	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71^{2} \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$1.350817753$	$24.25015266$	3.348483607	\( -\frac{107477168128}{5041} a^{3} - \frac{152866918400}{5041} a^{2} + \frac{629801447424}{5041} a + \frac{510363553792}{5041} \)	\( \bigl[0\) , \( -a^{2} + 3\) , \( 1\) , \( a^{2} + a - 6\) , \( 80 a^{3} - 140 a^{2} - 616 a + 1181\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(a^{2}+a-6\right){x}+80a^{3}-140a^{2}-616a+1181$
71.1-d1	71.1-d	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71 \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$0.077631787$	$821.9823011$	3.261437020	\( \frac{732083245280}{71} a^{3} - \frac{1278302446217}{71} a^{2} - \frac{5634987718502}{71} a + \frac{10793107895044}{71} \)	\( \bigl[a^{3} + 2 a^{2} - 6 a - 8\) , \( -2 a^{3} - 3 a^{2} + 11 a + 13\) , \( 2 a^{3} + 3 a^{2} - 12 a - 11\) , \( a^{3} - 2 a^{2} - 4 a + 8\) , \( 4 a^{3} - 3 a^{2} - 13 a\bigr] \)	${y}^2+\left(a^{3}+2a^{2}-6a-8\right){x}{y}+\left(2a^{3}+3a^{2}-12a-11\right){y}={x}^{3}+\left(-2a^{3}-3a^{2}+11a+13\right){x}^{2}+\left(a^{3}-2a^{2}-4a+8\right){x}+4a^{3}-3a^{2}-13a$
71.1-e1	71.1-e	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71^{3} \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 3 \)	$0.012087042$	$1375.479221$	2.549190457	\( \frac{40567729198}{357911} a^{3} - \frac{18573487223}{357911} a^{2} - \frac{289892232122}{357911} a + \frac{246977548593}{357911} \)	\( \bigl[a^{3} + 2 a^{2} - 6 a - 7\) , \( -a^{3} - 2 a^{2} + 6 a + 7\) , \( a + 1\) , \( -2 a^{3} - 6 a^{2} + 15 a + 16\) , \( 2 a^{3} - 3 a^{2} - 4 a\bigr] \)	${y}^2+\left(a^{3}+2a^{2}-6a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}-2a^{2}+6a+7\right){x}^{2}+\left(-2a^{3}-6a^{2}+15a+16\right){x}+2a^{3}-3a^{2}-4a$
71.1-f1	71.1-f	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71 \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$0.316585081$	$168.4566808$	2.725747549	\( \frac{732083245280}{71} a^{3} - \frac{1278302446217}{71} a^{2} - \frac{5634987718502}{71} a + \frac{10793107895044}{71} \)	\( \bigl[2 a^{3} + 3 a^{2} - 11 a - 12\) , \( -a^{3} - a^{2} + 6 a + 3\) , \( 1\) , \( -a^{3} + 8 a + 2\) , \( 6 a^{3} + 8 a^{2} - 38 a - 36\bigr] \)	${y}^2+\left(2a^{3}+3a^{2}-11a-12\right){x}{y}+{y}={x}^{3}+\left(-a^{3}-a^{2}+6a+3\right){x}^{2}+\left(-a^{3}+8a+2\right){x}+6a^{3}+8a^{2}-38a-36$
71.1-g1	71.1-g	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71 \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 1 \)	$0.021483562$	$2332.543676$	2.561197211	\( \frac{337705309521}{71} a^{3} + \frac{407692977510}{71} a^{2} - \frac{2139474325813}{71} a - \frac{1682987986020}{71} \)	\( \bigl[a^{3} + a^{2} - 5 a - 3\) , \( a^{2} + a - 5\) , \( 2 a^{3} + 3 a^{2} - 12 a - 12\) , \( -a^{3} - 3 a^{2} + 6 a + 15\) , \( 44 a^{3} + 61 a^{2} - 254 a - 211\bigr] \)	${y}^2+\left(a^{3}+a^{2}-5a-3\right){x}{y}+\left(2a^{3}+3a^{2}-12a-12\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(-a^{3}-3a^{2}+6a+15\right){x}+44a^{3}+61a^{2}-254a-211$
71.1-h1	71.1-h	$1$	$1$	4.4.6125.1	$4$	$[4, 0]$	71.1	\( 71 \)	\( 71^{2} \)	$11.91513$	$(-2a^3-2a^2+14a+5)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \)	$0.021909019$	$1115.169704$	2.497473303	\( -\frac{107477168128}{5041} a^{3} - \frac{152866918400}{5041} a^{2} + \frac{629801447424}{5041} a + \frac{510363553792}{5041} \)	\( \bigl[0\) , \( -a - 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -7 a^{3} - 10 a^{2} + 42 a + 34\) , \( 19 a^{3} + 26 a^{2} - 110 a - 89\bigr] \)	${y}^2+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a^{3}-10a^{2}+42a+34\right){x}+19a^{3}+26a^{2}-110a-89$

 

  *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.