Elliptic curves over \(\Q(\sqrt{-1}) \) of conductor 52650.3

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100000 over imaginary quadratic fields with absolute discriminant 4

Note: The completeness Only modular elliptic curves are included

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
52650.3-a1	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{4} \cdot 3^{12} \cdot 5^{7} \cdot 13^{3} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	\( 2^{3} \cdot 3^{2} \)	$2.508001076$	$0.321427698$	3.224564057	\( -\frac{1411302663595036}{34328125} a - \frac{1774751413484333}{137312500} \)	\( \bigl[1\) , \( -1\) , \( i + 1\) , \( -1454 i - 1985\) , \( -37787 i - 26892\bigr] \)	${y}^2+{x}{y}+\left(i+1\right){y}={x}^{3}-{x}^{2}+\left(-1454i-1985\right){x}-37787i-26892$
52650.3-a2	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{12} \cdot 3^{12} \cdot 5^{5} \cdot 13 \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	\( 2^{3} \)	$0.836000358$	$0.964283096$	3.224564057	\( -\frac{171697}{6500} a + \frac{2279159}{104000} \)	\( \bigl[1\) , \( -1\) , \( i + 1\) , \( -14 i - 5\) , \( -95 i - 108\bigr] \)	${y}^2+{x}{y}+\left(i+1\right){y}={x}^{3}-{x}^{2}+\left(-14i-5\right){x}-95i-108$
52650.3-a3	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2 \cdot 3^{12} \cdot 5^{25} \cdot 13^{3} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	\( 2^{5} \cdot 3^{2} \)	$2.508001076$	$0.080356924$	3.224564057	\( \frac{94290382838862669189021}{261902809143066406250} a + \frac{23228384730714798359947}{261902809143066406250} \)	\( \bigl[1\) , \( -1\) , \( i + 1\) , \( 4261 i + 1300\) , \( -222278 i - 102195\bigr] \)	${y}^2+{x}{y}+\left(i+1\right){y}={x}^{3}-{x}^{2}+\left(4261i+1300\right){x}-222278i-102195$
52650.3-a4	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{3} \cdot 3^{12} \cdot 5^{14} \cdot 13^{4} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	\( 2^{5} \)	$0.836000358$	$0.241070774$	3.224564057	\( -\frac{20122730162024161}{27891601562500} a + \frac{104798752060117927}{27891601562500} \)	\( \bigl[i\) , \( 1\) , \( i + 1\) , \( 976 i + 86\) , \( -4604 i - 7398\bigr] \)	${y}^2+i{x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(976i+86\right){x}-4604i-7398$
52650.3-a5	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{2} \cdot 3^{12} \cdot 5^{14} \cdot 13^{6} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/2\Z\oplus\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.1	$1$	\( 2^{7} \cdot 3^{2} \)	$1.254000538$	$0.160713849$	3.224564057	\( -\frac{12415547946147007137}{2356840332031250} a + \frac{5474429230691529908}{1178420166015625} \)	\( \bigl[i\) , \( 1\) , \( i + 1\) , \( -1544 i - 1984\) , \( 34330 i + 28026\bigr] \)	${y}^2+i{x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(-1544i-1984\right){x}+34330i+28026$
52650.3-a6	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2 \cdot 3^{12} \cdot 5^{10} \cdot 13^{12} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	\( 2^{5} \cdot 3^{2} \)	$2.508001076$	$0.080356924$	3.224564057	\( \frac{4240925829815707588031}{728065160077531250} a + \frac{3613304062782124177817}{728065160077531250} \)	\( \bigl[i\) , \( 1\) , \( i + 1\) , \( -8789 i - 5269\) , \( -373721 i + 26433\bigr] \)	${y}^2+i{x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(-8789i-5269\right){x}-373721i+26433$
52650.3-a7	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{6} \cdot 3^{12} \cdot 5^{10} \cdot 13^{2} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.2	$1$	\( 2^{7} \)	$0.418000179$	$0.482141548$	3.224564057	\( \frac{117057737097}{21125000} a + \frac{49160487287}{2640625} \)	\( \bigl[i\) , \( 1\) , \( i + 1\) , \( 346 i - 4\) , \( 1822 i + 1620\bigr] \)	${y}^2+i{x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(346i-4\right){x}+1822i+1620$
52650.3-a8	52650.3-a	$8$	$12$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{3} \cdot 3^{12} \cdot 5^{11} \cdot 13 \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	\( 2^{5} \)	$0.836000358$	$0.241070774$	3.224564057	\( \frac{4023422266102893}{20312500} a + \frac{5856979210600901}{20312500} \)	\( \bigl[1\) , \( -1\) , \( i + 1\) , \( 5476 i - 95\) , \( -114521 i - 107406\bigr] \)	${y}^2+{x}{y}+\left(i+1\right){y}={x}^{3}-{x}^{2}+\left(5476i-95\right){x}-114521i-107406$
52650.3-b1	52650.3-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{4} \cdot 3^{14} \cdot 5^{3} \cdot 13 \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{5} \)	$1$	$1.632144538$	3.264289077	\( -\frac{534692}{975} a + \frac{5321681}{3900} \)	\( \bigl[1\) , \( -1\) , \( i\) , \( 16 i + 3\) , \( 2 i - 17\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(16i+3\right){x}+2i-17$
52650.3-b2	52650.3-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2^{2} \cdot 3^{16} \cdot 5^{6} \cdot 13^{2} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	\( 2^{6} \)	$1$	$0.816072269$	3.264289077	\( \frac{1394971823}{1901250} a + \frac{203757348}{105625} \)	\( \bigl[1\) , \( -1\) , \( i\) , \( -74 i + 3\) , \( 56 i - 143\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(-74i+3\right){x}+56i-143$
52650.3-b3	52650.3-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2 \cdot 3^{20} \cdot 5^{9} \cdot 13 \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$4$	\( 2^{3} \)	$1$	$0.408036134$	3.264289077	\( -\frac{1328273569237}{30468750} a + \frac{23706220403743}{822656250} \)	\( \bigl[1\) , \( -1\) , \( i\) , \( -569 i - 132\) , \( -4750 i + 2269\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(-569i-132\right){x}-4750i+2269$
52650.3-b4	52650.3-b	$4$	$4$	\(\Q(\sqrt{-1}) \)	$2$	$[0, 1]$	52650.3	\( 2 \cdot 3^{4} \cdot 5^{2} \cdot 13 \)	\( 2 \cdot 3^{14} \cdot 5^{6} \cdot 13^{4} \)	$2.70719$	$(a+1), (-a-2), (2a+1), (-3a-2), (3)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{7} \)	$1$	$0.408036134$	3.264289077	\( \frac{190980574877449}{107103750} a + \frac{47588754812543}{107103750} \)	\( \bigl[1\) , \( -1\) , \( i\) , \( -1019 i + 138\) , \( 7238 i - 10619\bigr] \)	${y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(-1019i+138\right){x}+7238i-10619$

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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.