Elliptic curve search results

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
2916.1-b1	2916.1-b	$4$	$21$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	2916.1	\( 2^{2} \cdot 3^{6} \)	\( 2^{14} \cdot 3^{12} \)	$1.13736$	$(-2a+1), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.1[2], 7B.2.3	$1$	\( 3 \cdot 7 \)	$1$	$0.583485219$	1.572084960	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -1077\) , \( 13877\bigr] \)	${y}^2+{x}{y}={x}^{3}-{x}^{2}-1077{x}+13877$
13122.1-c1	13122.1-c	$4$	$21$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	13122.1	\( 2 \cdot 3^{8} \)	\( 2^{14} \cdot 3^{24} \)	$1.91280$	$(a+1), (3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.3	$1$	\( 2 \cdot 7 \)	$1$	$0.194495073$	2.722931025	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -9695\) , \( -364985\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-9695{x}-364985$
26244.2-l1	26244.2-l	$4$	$21$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	26244.2	\( 2^{2} \cdot 3^{8} \)	\( 2^{14} \cdot 3^{24} \)	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.1[2]	$1$	\( 7^{2} \)	$0.719702593$	$0.194495073$	10.36976045	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -9695\) , \( -364985\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-9695{x}-364985$
13122.5-e1	13122.5-e	$4$	$21$	\(\Q(\sqrt{-2}) \)	$2$	$[0, 1]$	13122.5	\( 2 \cdot 3^{8} \)	\( 2^{14} \cdot 3^{24} \)	$2.70510$	$(a), (-a-1), (a-1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2, 3, 7$	2Cn, 3B.1.2, 7B.2.3	$1$	\( 2 \cdot 7 \)	$1$	$0.194495073$	1.925402993	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -9695\) , \( -364985\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-9695{x}-364985$
26244.5-b1	26244.5-b	$4$	$21$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	26244.5	\( 2^{2} \cdot 3^{8} \)	\( 2^{14} \cdot 3^{24} \)	$3.77218$	$(-a), (a-1), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.3	$1$	\( 7 \)	$1$	$0.194495073$	0.820994594	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -9695\) , \( -364985\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-9695{x}-364985$
1458.1-p1	1458.1-p	$4$	$21$	\(\Q(\sqrt{3}) \)	$2$	$[2, 0]$	1458.1	\( 2 \cdot 3^{6} \)	\( 2^{14} \cdot 3^{12} \)	$1.91280$	$(a+1), (a)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.3	$1$	\( 2 \cdot 7 \)	$3.080173468$	$0.173617472$	4.322510075	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -1\) , \( a\) , \( 60322 a - 104487\) , \( 10598940 a - 18357907\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(60322a-104487\right){x}+10598940a-18357907$
8.1-a4	8.1-a	$4$	$21$	\(\Q(\zeta_{9})^+\)	$3$	$[3, 0]$	8.1	\( 2^{3} \)	\( - 2^{21} \)	$1.13736$	$(2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$1$	\( 7 \)	$1$	$0.375899741$	0.292366465	\( -\frac{189613868625}{128} \)	\( \bigl[a + 1\) , \( -a\) , \( a^{2} - 1\) , \( 17115 a^{2} - 5866 a - 49431\) , \( 1410206 a^{2} - 489214 a - 4061553\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}-a{x}^{2}+\left(17115a^{2}-5866a-49431\right){x}+1410206a^{2}-489214a-4061553$
64.1-a1	64.1-a	$4$	$21$	6.6.820125.1	$6$	$[6, 0]$	64.1	\( 2^{6} \)	\( 2^{42} \)	$114.44414$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$49$	\( 7 \)	$11.37381809$	$0.141300615$	3.65221	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -\frac{8}{19} a^{5} - \frac{1}{19} a^{4} + \frac{60}{19} a^{3} + \frac{49}{19} a^{2} + \frac{3}{19} a - \frac{45}{19}\) , \( \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{60}{19} a^{3} - \frac{49}{19} a^{2} - \frac{3}{19} a + \frac{45}{19}\) , \( -\frac{2}{19} a^{5} - \frac{5}{19} a^{4} + \frac{15}{19} a^{3} + \frac{55}{19} a^{2} + \frac{15}{19} a - \frac{2277}{19}\) , \( \frac{314}{19} a^{5} + \frac{44}{19} a^{4} - \frac{2355}{19} a^{3} - \frac{1966}{19} a^{2} - \frac{132}{19} a - \frac{7995}{19}\bigr] \)	${y}^2+{x}{y}+\left(\frac{8}{19}a^{5}+\frac{1}{19}a^{4}-\frac{60}{19}a^{3}-\frac{49}{19}a^{2}-\frac{3}{19}a+\frac{45}{19}\right){y}={x}^{3}+\left(-\frac{8}{19}a^{5}-\frac{1}{19}a^{4}+\frac{60}{19}a^{3}+\frac{49}{19}a^{2}+\frac{3}{19}a-\frac{45}{19}\right){x}^{2}+\left(-\frac{2}{19}a^{5}-\frac{5}{19}a^{4}+\frac{15}{19}a^{3}+\frac{55}{19}a^{2}+\frac{15}{19}a-\frac{2277}{19}\right){x}+\frac{314}{19}a^{5}+\frac{44}{19}a^{4}-\frac{2355}{19}a^{3}-\frac{1966}{19}a^{2}-\frac{132}{19}a-\frac{7995}{19}$
8.1-b1	8.1-b	$4$	$21$	\(\Q(\zeta_{36})^+\)	$6$	$[6, 0]$	8.1	\( 2^{3} \)	\( 2^{42} \)	$119.27013$	$(-a^4-a^3+3a^2+2a-1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$49$	\( 2 \cdot 7 \)	$3.080173468$	$0.141300615$	1.59609	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( a^{4} - 6 a^{2} + 5\) , \( a^{4} - 4 a^{2} + 3\) , \( -118\) , \( -40 a^{4} + 239 a^{2} - 713\bigr] \)	${y}^2+{x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(a^{4}-6a^{2}+5\right){x}^{2}-118{x}-40a^{4}+239a^{2}-713$
64.1-b1	64.1-b	$4$	$21$	6.6.1292517.1	$6$	$[6, 0]$	64.1	\( 2^{6} \)	\( 2^{42} \)	$143.67185$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$49$	\( 7 \)	$15.86870131$	$0.141300615$	4.05894	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 6 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 6 a^{2} + 4 a + 3\) , \( -a^{5} - a^{4} + 6 a^{3} + 6 a^{2} - 4 a - 120\) , \( 39 a^{5} - 40 a^{4} - 234 a^{3} + 161 a^{2} + 235 a - 555\bigr] \)	${y}^2+{x}{y}+\left(a^{5}+a^{4}-6a^{3}-6a^{2}+4a+3\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-6a+1\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+6a^{2}-4a-120\right){x}+39a^{5}-40a^{4}-234a^{3}+161a^{2}+235a-555$
64.1-d1	64.1-d	$4$	$21$	6.6.1397493.1	$6$	$[6, 0]$	64.1	\( 2^{6} \)	\( 2^{42} \)	$149.39235$	$(2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$49$	\( 7 \)	$8.578636127$	$0.141300615$	2.11024	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( -a^{5} + 3 a^{4} + 4 a^{3} - 11 a^{2} - 6 a + 4\) , \( a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 7 a^{2} - 2 a - 117\) , \( 39 a^{5} - 117 a^{4} - 157 a^{3} + 431 a^{2} + 236 a - 672\bigr] \)	${y}^2+{x}{y}+\left(a^{5}-3a^{4}-2a^{3}+7a^{2}+2a\right){y}={x}^{3}+\left(-a^{5}+3a^{4}+4a^{3}-11a^{2}-6a+4\right){x}^{2}+\left(-a^{5}+3a^{4}+2a^{3}-7a^{2}-2a-117\right){x}+39a^{5}-117a^{4}-157a^{3}+431a^{2}+236a-672$
64.1-g1	64.1-g	$4$	$21$	6.6.1528713.1	$6$	$[6, 0]$	64.1	\( 2^{6} \)	\( 2^{42} \)	$156.24874$	$(2a^5-6a^4-5a^3+12a^2+a-3), (-a^5+3a^4+3a^3-6a^2-2a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3, 7$	3B.1.2, 7B.1.3	$441$	\( 7^{2} \)	$1$	$0.141300615$	2.46954	\( -\frac{189613868625}{128} \)	\( \bigl[1\) , \( a^{5} - 5 a^{4} + 3 a^{3} + 11 a^{2} - 6 a - 5\) , \( a^{5} - 3 a^{4} - 3 a^{3} + 7 a^{2} + 2 a - 1\) , \( -118\) , \( -40 a^{5} + 199 a^{4} - 117 a^{3} - 438 a^{2} + 236 a - 316\bigr] \)	${y}^2+{x}{y}+\left(a^{5}-3a^{4}-3a^{3}+7a^{2}+2a-1\right){y}={x}^{3}+\left(a^{5}-5a^{4}+3a^{3}+11a^{2}-6a-5\right){x}^{2}-118{x}-40a^{5}+199a^{4}-117a^{3}-438a^{2}+236a-316$

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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.