Elliptic curves over \(\Q(\sqrt{21}) \) of conductor 847.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over real quadratic fields with discriminant 497

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
847.1-a1	847.1-a	$2$	$2$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{6} \cdot 11^{4} \)	$2.20912$	$(a+3), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{2} \)	$1.480301811$	$3.210187340$	2.073962963	\( \frac{4657463}{41503} \)	\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 18 a + 34\) , \( -195 a - 351\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(18a+34\right){x}-195a-351$
847.1-a2	847.1-a	$2$	$2$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{12} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2 \)	$2.960603622$	$3.210187340$	2.073962963	\( \frac{15124197817}{1294139} \)	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 260 a - 717\) , \( 3155 a - 8805\bigr] \)	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(260a-717\right){x}+3155a-8805$
847.1-b1	847.1-b	$1$	$1$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓				$1$	\( 2 \)	$0.098027979$	$10.23860076$	0.876074426	\( \frac{884736}{539} \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 2\) , \( 0\bigr] \)	${y}^2+{y}={x}^{3}+2{x}$
847.1-c1	847.1-c	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2^{2} \)	$1$	$21.59210152$	2.094125706	\( -\frac{78843215872}{539} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( -89\) , \( 295\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}-89{x}+295$
847.1-c2	847.1-c	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{12} \cdot 11^{6} \)	$2.20912$	$(a+3), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$2.399122391$	2.094125706	\( -\frac{13278380032}{156590819} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( -49\) , \( 600\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}-49{x}+600$
847.1-c3	847.1-c	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{18} \)	$2.20912$	$(a+3), (11)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$0.266569154$	2.094125706	\( \frac{9463555063808}{115539436859} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( 441\) , \( -15815\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}+441{x}-15815$
847.1-d1	847.1-d	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2^{2} \)	$1$	$0.602881548$	0.526238158	\( -\frac{78843215872}{539} \)	\( \bigl[0\) , \( a\) , \( 1\) , \( 447 a - 1249\) , \( 7532 a - 21030\bigr] \)	${y}^2+{y}={x}^{3}+a{x}^{2}+\left(447a-1249\right){x}+7532a-21030$
847.1-d2	847.1-d	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{12} \cdot 11^{6} \)	$2.20912$	$(a+3), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$0.602881548$	0.526238158	\( -\frac{13278380032}{156590819} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -247 a - 442\) , \( -14652 a - 26253\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-247a-442\right){x}-14652a-26253$
847.1-d3	847.1-d	$3$	$9$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{18} \)	$2.20912$	$(a+3), (11)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$0.602881548$	0.526238158	\( \frac{9463555063808}{115539436859} \)	\( \bigl[0\) , \( a\) , \( 1\) , \( -2203 a + 6171\) , \( -381758 a + 1065760\bigr] \)	${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-2203a+6171\right){x}-381758a+1065760$
847.1-e1	847.1-e	$2$	$2$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{6} \cdot 11^{4} \)	$2.20912$	$(a+3), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{2} \)	$0.833846915$	$7.031061357$	2.558748273	\( \frac{4657463}{41503} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( 4\) , \( 11\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}+4{x}+11$
847.1-e2	847.1-e	$2$	$2$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{12} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2 \)	$1.667693830$	$7.031061357$	2.558748273	\( \frac{15124197817}{1294139} \)	\( \bigl[1\) , \( 1\) , \( 0\) , \( -51\) , \( 110\bigr] \)	${y}^2+{x}{y}={x}^{3}+{x}^{2}-51{x}+110$
847.1-f1	847.1-f	$1$	$1$	\(\Q(\sqrt{21}) \)	$2$	$[2, 0]$	847.1	\( 7 \cdot 11^{2} \)	\( 7^{4} \cdot 11^{2} \)	$2.20912$	$(a+3), (11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓				$1$	\( 2 \)	$0.563972113$	$9.086092621$	4.472858235	\( \frac{884736}{539} \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 10 a + 18\) , \( -6 a - 11\bigr] \)	${y}^2+{y}={x}^{3}+\left(10a+18\right){x}-6a-11$

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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.