Elliptic curve search results

Results (10 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
4.1-a1	4.1-a	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	4.1	\( 2^{2} \)	\( 2^{4} \)	$0.95409$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$3, 7$	3B, 7B.6.3	$1$	\( 1 \)	$1$	$7.662151750$	1.014876791	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 71395 a - 305260\) , \( -20071186 a + 85802856\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(71395a-305260\right){x}-20071186a+85802856$
4.1-d1	4.1-d	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	4.1	\( 2^{2} \)	\( 2^{4} \)	$0.95409$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$3, 7$	3B, 7B.1.3	$49$	\( 3 \)	$1$	$0.077866687$	1.516113114	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[1\) , \( a\) , \( a\) , \( -642629 a - 2104558\) , \( -545795629 a - 1787435506\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-642629a-2104558\right){x}-545795629a-1787435506$
36.1-b1	36.1-b	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	36.1	\( 2^{2} \cdot 3^{2} \)	\( 2^{4} \cdot 3^{6} \)	$1.65254$	$(a-4), (a+3), (4a+13)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B.1.2, 7B.6.3	$49$	\( 1 \)	$1$	$0.257471977$	1.671046829	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[1\) , \( -1\) , \( a\) , \( 64690450 a - 276546321\) , \( -550295059294 a + 2352465823751\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(64690450a-276546321\right){x}-550295059294a+2352465823751$
36.1-f1	36.1-f	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	36.1	\( 2^{2} \cdot 3^{2} \)	\( 2^{4} \cdot 3^{6} \)	$1.65254$	$(a-4), (a+3), (4a+13)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B.1.1, 7B.6.3	$49$	\( 3 \)	$1$	$0.772415932$	1.671046829	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -5675 a - 23939\) , \( 560829 a + 1687167\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5675a-23939\right){x}+560829a+1687167$
128.5-c1	128.5-c	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	128.5	\( 2^{7} \)	\( 2^{22} \)	$2.26923$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$49$	\( 2 \)	$1$	$0.473006225$	6.139818101	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[a\) , \( a\) , \( a\) , \( 115785 a - 497049\) , \( 41797219 a - 178648906\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(115785a-497049\right){x}+41797219a-178648906$
128.5-j1	128.5-j	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	128.5	\( 2^{7} \)	\( 2^{22} \)	$2.26923$	$(a-4), (a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$1$	\( 2^{2} \)	$9.903074376$	$0.157668741$	1.654505450	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -25303280 a - 82866156\) , \( -134641581530 a - 440940033556\bigr] \)	${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-25303280a-82866156\right){x}-134641581530a-440940033556$
128.6-c1	128.6-c	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	128.6	\( 2^{7} \)	\( 2^{22} \)	$2.26923$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$49$	\( 2 \cdot 3 \)	$1$	$0.157668741$	6.139818101	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 2811452 a - 12018824\) , \( -4982304577 a + 21298939255\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2811452a-12018824\right){x}-4982304577a+21298939255$
128.6-j1	128.6-j	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	128.6	\( 2^{7} \)	\( 2^{22} \)	$2.26923$	$(a-4), (a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$1$	\( 2^{2} \)	$3.301024792$	$0.473006225$	1.654505450	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -1044502 a - 3420890\) , \( 1124767860 a + 3683520340\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1044502a-3420890\right){x}+1124767860a+3683520340$
256.1-a1	256.1-a	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{28} \)	$2.69858$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$1$	\( 2^{4} \)	$1$	$1.915537937$	4.059507167	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -10282063 a - 33672998\) , \( 34886965210 a + 114251923320\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10282063a-33672998\right){x}+34886965210a+114251923320$
256.1-r1	256.1-r	$4$	$21$	\(\Q(\sqrt{57}) \)	$2$	$[2, 0]$	256.1	\( 2^{8} \)	\( 2^{28} \)	$2.69858$	$(a-4), (a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 7$	3B, 7B.6.3	$49$	\( 2^{2} \)	$1$	$0.019466671$	0.505371038	\( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 1142337 a - 4884238\) , \( 1290582430 a - 5517144100\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(1142337a-4884238\right){x}+1290582430a-5517144100$

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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.