Elliptic curves over Q(−1)\Q(\sqrt{-1}) of conductor 57800.4

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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
57800.4-a1	57800.4-a	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{4} \cdot 17^{3}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$2.137996157$	2.137996157	$-\frac{71702}{125} a + \frac{470336}{125}$	$\bigl[i + 1$ , $i - 1$ , $i + 1$ , $8 i + 7$ , $-3 i + 5\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i-1\right){x}^{2}+\left(8i+7\right){x}-3i+5$
57800.4-a2	57800.4-a	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{8} \cdot 17^{3}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$1$	$1.068998078$	2.137996157	$\frac{70930131}{15625} a + \frac{299889467}{15625}$	$\bigl[i + 1$ , $i - 1$ , $i + 1$ , $58 i + 37$ , $7 i - 261\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i-1\right){x}^{2}+\left(58i+37\right){x}+7i-261$
57800.4-b1	57800.4-b	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{4} \cdot 17^{9}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2} \cdot 3$	$1$	$0.518540234$	1.555620703	$-\frac{71702}{125} a + \frac{470336}{125}$	$\bigl[i + 1$ , $-i - 1$ , $i + 1$ , $-210 i - 31$ , $849 i - 449\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-210i-31\right){x}+849i-449$
57800.4-b2	57800.4-b	$2$	$2$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{8} \cdot 17^{9}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$1$	$0.259270117$	1.555620703	$\frac{70930131}{15625} a + \frac{299889467}{15625}$	$\bigl[i + 1$ , $-i - 1$ , $i + 1$ , $-1200 i - 81$ , $-11373 i + 9653\bigr]$	${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-1200i-81\right){x}-11373i+9653$
57800.4-c1	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{4} \cdot 17^{7}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{6}$	$2.303606929$	$0.217024174$	3.999507146	$-\frac{2226135040016}{425} a - \frac{4178441913604}{425}$	$\bigl[i + 1$ , $i + 1$ , $0$ , $-6933 i + 6265$ , $109144 i + 335983\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-6933i+6265\right){x}+109144i+335983$
57800.4-c2	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{4} \cdot 5^{8} \cdot 17^{8}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{6}$	$1.151803464$	$0.434048349$	3.999507146	$\frac{18495673728}{180625} a - \frac{897072368}{36125}$	$\bigl[i + 1$ , $i + 1$ , $0$ , $-433 i + 390$ , $1769 i + 5508\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-433i+390\right){x}+1769i+5508$
57800.4-c3	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{5} \cdot 17^{14}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$4.607213858$	$0.108512087$	3.999507146	$-\frac{624467745025896476}{4359848400625} a - \frac{74500491067519382}{4359848400625}$	$\bigl[i + 1$ , $i + 1$ , $0$ , $5887 i + 7905$ , $-212070 i + 307155\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(5887i+7905\right){x}-212070i+307155$
57800.4-c4	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{10} \cdot 17^{10}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{6}$	$2.303606929$	$0.217024174$	3.999507146	$\frac{1142278337424}{32625390625} a + \frac{4669682943668}{32625390625}$	$\bigl[i + 1$ , $i + 1$ , $0$ , $-13 i + 455$ , $-1940 i + 12245\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-13i+455\right){x}-1940i+12245$
57800.4-c5	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{10} \cdot 17^{7}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{7}$	$0.575901732$	$0.434048349$	3.999507146	$-\frac{2845155328}{6640625} a + \frac{8254109696}{6640625}$	$\bigl[0$ , $-i$ , $0$ , $216 i - 77$ , $261 i + 979\bigr]$	${y}^2={x}^{3}-i{x}^{2}+\left(216i-77\right){x}+261i+979$
57800.4-c6	57800.4-c	$6$	$8$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{17} \cdot 17^{8}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$4.607213858$	$0.108512087$	3.999507146	$-\frac{54765023102363044}{44097900390625} a + \frac{449923792854324742}{44097900390625}$	$\bigl[i + 1$ , $i + 1$ , $0$ , $807 i - 5955$ , $-37466 i + 157543\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(807i-5955\right){x}-37466i+157543$
57800.4-d1	57800.4-d	$1$	$1$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{4} \cdot 5^{2} \cdot 17^{2}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2$	$0.207456378$	$5.029645956$	4.173728546	$-2048 a - \frac{6144}{5}$	$\bigl[0$ , $-i$ , $i + 1$ , $i - 2$ , $-i + 1\bigr]$	${y}^2+\left(i+1\right){y}={x}^{3}-i{x}^{2}+\left(i-2\right){x}-i+1$
57800.4-e1	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{10} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{7}$	$1$	$0.363426171$	2.907409369	$-\frac{35999730234}{390625} a - \frac{51700389912}{390625}$	$\bigl[i + 1$ , $i$ , $0$ , $796 i + 408$ , $936 i + 9924\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(796i+408\right){x}+936i+9924$
57800.4-e2	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{10} \cdot 5^{10} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$4$	$2^{5}$	$1$	$0.363426171$	2.907409369	$\frac{35999730234}{390625} a - \frac{51700389912}{390625}$	$\bigl[i + 1$ , $i$ , $0$ , $-104 i + 888$ , $10640 i + 1820\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-104i+888\right){x}+10640i+1820$
57800.4-e3	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{8} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{6}$	$1$	$0.726852342$	2.907409369	$\frac{237276}{625}$	$\bigl[i + 1$ , $i$ , $0$ , $26 i + 48$ , $224 i + 208\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(26i+48\right){x}+224i+208$
57800.4-e4	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{17} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$16$	$2^{2}$	$1$	$0.181713085$	2.907409369	$-\frac{22845545233191}{152587890625} a + \frac{135893651813613}{152587890625}$	$\bigl[i + 1$ , $i$ , $0$ , $-634 i + 978$ , $9964 i - 11152\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-634i+978\right){x}+9964i-11152$
57800.4-e5	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{17} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{6}$	$1$	$0.181713085$	2.907409369	$\frac{22845545233191}{152587890625} a + \frac{135893651813613}{152587890625}$	$\bigl[i + 1$ , $i$ , $0$ , $1166 i + 18$ , $-12356 i + 7688\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(1166i+18\right){x}-12356i+7688$
57800.4-e6	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{4} \cdot 5^{4} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$1.453704684$	2.907409369	$\frac{148176}{25}$	$\bigl[i + 1$ , $i$ , $0$ , $-14 i - 27$ , $22 i + 46\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-14i-27\right){x}+22i+46$
57800.4-e7	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{2} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$1$	$1.453704684$	2.907409369	$\frac{55296}{5}$	$\bigl[0$ , $0$ , $0$ , $16 i + 30$ , $52 i - 47\bigr]$	${y}^2={x}^{3}+\left(16i+30\right){x}+52i-47$
57800.4-e8	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{2} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$4$	$2^{2}$	$1$	$0.726852342$	2.907409369	$\frac{132304644}{5}$	$\bigl[i + 1$ , $i$ , $0$ , $-214 i - 402$ , $2302 i + 2876\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-214i-402\right){x}+2302i+2876$
57800.4-e9	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{5} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$16$	$2^{2}$	$1$	$0.181713085$	2.907409369	$-\frac{15332659200009}{625} a + \frac{5763174879987}{625}$	$\bigl[i + 1$ , $i$ , $0$ , $-1654 i + 14238$ , $657780 i + 114840\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-1654i+14238\right){x}+657780i+114840$
57800.4-e10	57800.4-e	$10$	$16$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{11} \cdot 5^{5} \cdot 17^{6}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$4$	$2^{4}$	$1$	$0.181713085$	2.907409369	$\frac{15332659200009}{625} a + \frac{5763174879987}{625}$	$\bigl[i + 1$ , $i$ , $0$ , $12746 i + 6558$ , $51156 i + 652464\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(12746i+6558\right){x}+51156i+652464$
57800.4-f1	57800.4-f	$1$	$1$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{4} \cdot 5^{2} \cdot 17^{8}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2$	$1$	$1.219868325$	2.439736651	$-2048 a - \frac{6144}{5}$	$\bigl[0$ , $-i - 1$ , $i + 1$ , $-i + 33$ , $75 i + 14\bigr]$	${y}^2+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-i+33\right){x}+75i+14$
57800.4-g1	57800.4-g	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{4} \cdot 5^{6} \cdot 17^{8}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{6}$	$1$	$0.738190738$	2.952762954	$\frac{33574464}{180625} a + \frac{283128848}{180625}$	$\bigl[i + 1$ , $-i + 1$ , $0$ , $83 i + 12$ , $-91 i + 28\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(83i+12\right){x}-91i+28$
57800.4-g2	57800.4-g	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{6} \cdot 17^{7}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{6}$	$1$	$0.738190738$	2.952762954	$-\frac{2306048}{10625} a + \frac{19982336}{10625}$	$\bigl[0$ , $-i + 1$ , $0$ , $86 i + 18$ , $16 i - 85\bigr]$	${y}^2={x}^{3}+\left(-i+1\right){x}^{2}+\left(86i+18\right){x}+16i-85$
57800.4-g3	57800.4-g	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{9} \cdot 17^{7}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{7}$	$1$	$0.369095369$	2.952762954	$-\frac{932738084712}{6640625} a + \frac{486943284916}{6640625}$	$\bigl[i + 1$ , $-i + 1$ , $0$ , $813 i + 297$ , $2696 i + 9697\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(813i+297\right){x}+2696i+9697$
57800.4-g4	57800.4-g	$4$	$4$	$\Q(\sqrt{-1})$	$2$	$[0, 1]$	57800.4	$2^{3} \cdot 5^{2} \cdot 17^{2}$	$2^{8} \cdot 5^{3} \cdot 17^{10}$	$2.77109$	$(a+1), (-a-2), (2a+1), (a+4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{5}$	$1$	$0.369095369$	2.952762954	$\frac{932967242152}{2088025} a + \frac{369264775804}{2088025}$	$\bigl[i + 1$ , $-i + 1$ , $0$ , $1033 i - 13$ , $-8876 i - 8677\bigr]$	${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(1033i-13\right){x}-8876i-8677$

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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.