Elliptic curves over \(\Q(\sqrt{-42}) \) of conductor 21.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100 over imaginary quadratic fields with absolute discriminant 168

Note: The completeness Only modular elliptic curves are included

Results (24 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
21.1-a1	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{2} \cdot 7^{16} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{5} \)	$2.638508823$	$0.862076929$	5.615648435	\( -\frac{4354703137}{17294403} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -85\) , \( 2123\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-85{x}+2123$
21.1-a2	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{4} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{3} \)	$1.319254411$	$6.896615437$	5.615648435	\( \frac{103823}{63} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( 55\) , \( -33\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+55{x}-33$
21.1-a3	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{8} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{5} \)	$2.638508823$	$3.448307718$	5.615648435	\( \frac{7189057}{3969} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( 35\) , \( 35\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+35{x}+35$
21.1-a4	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{16} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{5} \)	$1.319254411$	$1.724153859$	5.615648435	\( \frac{6570725617}{45927} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -105\) , \( -273\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-105{x}-273$
21.1-a5	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{4} \cdot 7^{8} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{5} \)	$5.277017646$	$1.724153859$	5.615648435	\( \frac{13027640977}{21609} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -145\) , \( 1655\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-145{x}+1655$
21.1-a6	21.1-a	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 2^{12} \cdot 3^{2} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{3} \)	$10.55403529$	$0.862076929$	5.615648435	\( \frac{53297461115137}{147} \)	\( \bigl[a\) , \( 1\) , \( a\) , \( -3085\) , \( 77507\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3+{x}^2-3085{x}+77507$
21.1-b1	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{16} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1$	$0.862076929$	0.532085432	\( -\frac{4354703137}{17294403} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \)	${y}^2+{x}{y}={x}^3-34{x}-217$
21.1-b2	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{3} \)	$1$	$6.896615437$	0.532085432	\( \frac{103823}{63} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \)	${y}^2+{x}{y}={x}^3+{x}$
21.1-b3	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{8} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$1$	$3.448307718$	0.532085432	\( \frac{7189057}{3969} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \)	${y}^2+{x}{y}={x}^3-4{x}-1$
21.1-b4	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{16} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{5} \)	$1$	$1.724153859$	0.532085432	\( \frac{6570725617}{45927} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \)	${y}^2+{x}{y}={x}^3-39{x}+90$
21.1-b5	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{8} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{3} \)	$1$	$1.724153859$	0.532085432	\( \frac{13027640977}{21609} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \)	${y}^2+{x}{y}={x}^3-49{x}-136$
21.1-b6	21.1-b	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1$	$0.862076929$	0.532085432	\( \frac{53297461115137}{147} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \)	${y}^2+{x}{y}={x}^3-784{x}-8515$
21.1-c1	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{28} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{5} \)	$1$	$0.862076929$	1.064170865	\( -\frac{4354703137}{17294403} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -1667\) , \( 72764\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2-1667{x}+72764$
21.1-c2	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{14} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1$	$6.896615437$	1.064170865	\( \frac{103823}{63} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( 48\) , \( 48\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2+48{x}+48$
21.1-c3	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{8} \cdot 7^{16} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{3} \)	$1$	$3.448307718$	1.064170865	\( \frac{7189057}{3969} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -197\) , \( 146\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2-197{x}+146$
21.1-c4	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{16} \cdot 7^{14} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1$	$1.724153859$	1.064170865	\( \frac{6570725617}{45927} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -1912\) , \( -32782\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2-1912{x}-32782$
21.1-c5	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{4} \cdot 7^{20} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{4} \)	$1$	$1.724153859$	1.064170865	\( \frac{13027640977}{21609} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -2402\) , \( 44246\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2-2402{x}+44246$
21.1-c6	21.1-c	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{2} \cdot 7^{16} \)	$2.47941$	$(3,a), (7,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{3} \)	$1$	$0.862076929$	1.064170865	\( \frac{53297461115137}{147} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -38417\) , \( 2882228\bigr] \)	${y}^2+{x}{y}+{y}={x}^3+{x}^2-38417{x}+2882228$
21.1-d1	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{14} \cdot 7^{16} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$13.74319437$	$0.862076929$	3.656276762	\( -\frac{4354703137}{17294403} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -306\) , \( 5859\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-306{x}+5859$
21.1-d2	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{16} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{2} \)	$1.717899296$	$6.896615437$	3.656276762	\( \frac{103823}{63} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( 9\) , \( 0\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2+9{x}$
21.1-d3	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{20} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$3.435798593$	$3.448307718$	3.656276762	\( \frac{7189057}{3969} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -36\) , \( 27\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-36{x}+27$
21.1-d4	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{28} \cdot 7^{2} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$1.717899296$	$1.724153859$	3.656276762	\( \frac{6570725617}{45927} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -351\) , \( -2430\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-351{x}-2430$
21.1-d5	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{16} \cdot 7^{8} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$6.871597187$	$1.724153859$	3.656276762	\( \frac{13027640977}{21609} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -441\) , \( 3672\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-441{x}+3672$
21.1-d6	21.1-d	$6$	$8$	\(\Q(\sqrt{-42}) \)	$2$	$[0, 1]$	21.1	\( 3 \cdot 7 \)	\( 3^{14} \cdot 7^{4} \)	$2.47941$	$(3,a), (7,a)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	\( 2^{2} \)	$13.74319437$	$0.862076929$	3.656276762	\( \frac{53297461115137}{147} \)	\( \bigl[1\) , \( -1\) , \( 0\) , \( -7056\) , \( 229905\bigr] \)	${y}^2+{x}{y}={x}^3-{x}^2-7056{x}+229905$

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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.