Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 22188.2

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 150000 over imaginary quadratic fields with absolute discriminant 3

Note: The completeness Only modular elliptic curves are included

Results (15 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
22188.2-a1	22188.2-a	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{12} \cdot 3^{2} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓				$1$	\( 2^{2} \cdot 3 \)	$0.017084761$	$3.102903120$	2.938243290	\( \frac{1685159}{8256} \)	\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 3 a - 4\) , \( -6\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(3a-4\right){x}-6$
22188.2-b1	22188.2-b	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{38} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓				$4$	\( 2^{2} \)	$1$	$0.076317390$	1.409979710	\( -\frac{9500554530751882177}{199908972324} \)	\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -44123 a + 44123\) , \( 3593277\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-44123a+44123\right){x}+3593277$
22188.2-c1	22188.2-c	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{14} \cdot 3^{28} \cdot 43^{4} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{3} \cdot 7 \)	$1$	$0.136046132$	2.199295590	\( -\frac{230042158153417}{1131994839168} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( -1276\) , \( 53584\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}-1276{x}+53584$
22188.2-c2	22188.2-c	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{28} \cdot 3^{14} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{2} \cdot 7 \)	$1$	$0.272092264$	2.199295590	\( \frac{778510269523657}{1540767744} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( -1916\) , \( 31440\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}-1916{x}+31440$
22188.2-d1	22188.2-d	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{6} \cdot 3^{8} \cdot 43^{8} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{3} \cdot 3 \)	$1$	$0.373008811$	2.584280852	\( -\frac{3107661785857}{2215383048} \)	\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 303 a\) , \( -3175\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+303a{x}-3175$
22188.2-d2	22188.2-d	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{24} \cdot 3^{2} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{3} \cdot 3 \)	$1$	$1.492035245$	2.584280852	\( \frac{1532808577}{528384} \)	\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 23 a\) , \( -39\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+23a{x}-39$
22188.2-d3	22188.2-d	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{12} \cdot 3^{4} \cdot 43^{4} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{4} \cdot 3 \)	$1$	$0.746017622$	2.584280852	\( \frac{4502751117697}{1065024} \)	\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 343 a\) , \( -2599\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+343a{x}-2599$
22188.2-d4	22188.2-d	$4$	$4$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{6} \cdot 3^{2} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$4$	\( 2 \cdot 3 \)	$1$	$0.373008811$	2.584280852	\( \frac{18440127492397057}{1032} \)	\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 5503 a\) , \( -159463\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+5503a{x}-159463$
22188.2-e1	22188.2-e	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{8} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{4} \)	$0.031712238$	$2.705889273$	3.170708154	\( \frac{224434}{129} a + \frac{31132009}{13932} \)	\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -6 a + 8\) , \( 4 a\bigr] \)	${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+8\right){x}+4a$
22188.2-f1	22188.2-f	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{8} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2^{4} \)	$0.031712238$	$2.705889273$	3.170708154	\( -\frac{224434}{129} a + \frac{55370881}{13932} \)	\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -8 a + 7\) , \( -6 a + 12\bigr] \)	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a+7\right){x}-6a+12$
22188.2-g1	22188.2-g	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{10} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓				$1$	\( 2^{2} \cdot 5 \)	$0.033607720$	$2.132623693$	3.310416574	\( -\frac{338608873}{41796} \)	\( \bigl[a\) , \( 0\) , \( 1\) , \( 14 a\) , \( 22\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+14a{x}+22$
22188.2-h1	22188.2-h	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{2} \cdot 3^{4} \cdot 43^{4} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{4} \)	$0.159052025$	$2.425033706$	3.563004244	\( \frac{56181887}{33282} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( -8 a\) , \( 2\bigr] \)	${y}^2+a{x}{y}={x}^{3}-8a{x}+2$
22188.2-h2	22188.2-h	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{2} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{2} \)	$0.318104051$	$4.850067413$	3.563004244	\( \frac{912673}{516} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 2 a\) , \( 0\bigr] \)	${y}^2+a{x}{y}={x}^{3}+2a{x}$
22188.2-i1	22188.2-i	$2$	$7$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{4} \cdot 3^{2} \cdot 43^{14} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$7$	7B.1.3	$1$	\( 2^{2} \cdot 7^{2} \)	$0.144128547$	$0.058551767$	3.819842511	\( -\frac{23769846831649063249}{3261823333284} \)	\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -59901 a + 59901\) , \( -5648523\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-59901a+59901\right){x}-5648523$
22188.2-i2	22188.2-i	$2$	$7$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	22188.2	\( 2^{2} \cdot 3 \cdot 43^{2} \)	\( 2^{28} \cdot 3^{14} \cdot 43^{2} \)	$1.88899$	$(-2a+1), (-7a+1), (7a-6), (2)$	$1$	$\Z/7\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$7$	7B.1.1	$1$	\( 2^{2} \cdot 7^{2} \)	$1.008899832$	$0.409862375$	3.819842511	\( \frac{444369620591}{1540767744} \)	\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 159 a - 159\) , \( 1737\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(159a-159\right){x}+1737$

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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.