Elliptic curves over \(\Q(\sqrt{-7}) \) of conductor 16900.2

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 50000 over imaginary quadratic fields with absolute discriminant 7

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
16900.2-a1	16900.2-a	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{4} \cdot 5^{12} \cdot 13^{4} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{4} \cdot 3 \)	$0.585232076$	$0.946074749$	1.116100444	\( -\frac{217081801}{10562500} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -13\) , \( 156\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-13{x}+156$
16900.2-a2	16900.2-a	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{12} \cdot 5^{4} \cdot 13^{12} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{4} \cdot 3 \)	$0.195077358$	$0.315358249$	1.116100444	\( \frac{157376536199}{7722894400} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( 112\) , \( -4194\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+112{x}-4194$
16900.2-a3	16900.2-a	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{24} \cdot 5^{2} \cdot 13^{6} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{2} \cdot 3 \)	$0.390154717$	$0.630716499$	1.116100444	\( \frac{988345570681}{44994560} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -208\) , \( -1122\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-208{x}-1122$
16900.2-a4	16900.2-a	$4$	$6$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{8} \cdot 5^{6} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3 \)	$1.170464153$	$1.892149498$	1.116100444	\( \frac{3803721481}{26000} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -33\) , \( 68\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-33{x}+68$
16900.2-b1	16900.2-b	$1$	$1$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{10} \cdot 5^{4} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \cdot 5 \)	$0.106009335$	$2.549984976$	4.086887690	\( -\frac{3542491}{260} a + \frac{185279897}{10400} \)	\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 6 a + 6\) , \( -4 a + 12\bigr] \)	${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6a+6\right){x}-4a+12$
16900.2-c1	16900.2-c	$1$	$1$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{10} \cdot 5^{4} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 2 \cdot 5 \)	$0.106009335$	$2.549984976$	4.086887690	\( \frac{3542491}{260} a + \frac{43580257}{10400} \)	\( \bigl[1\) , \( -a\) , \( 0\) , \( -6 a + 12\) , \( 4 a + 8\bigr] \)	${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-6a+12\right){x}+4a+8$
16900.2-d1	16900.2-d	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{8} \cdot 5^{20} \cdot 13^{4} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{6} \cdot 5 \)	$0.264724691$	$0.244864813$	7.840103432	\( -\frac{48743122863889}{26406250000} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -761\) , \( -11561\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-761{x}-11561$
16900.2-d2	16900.2-d	$2$	$2$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{16} \cdot 5^{10} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{6} \cdot 5 \)	$0.132362345$	$0.489729627$	7.840103432	\( \frac{65787589563409}{10400000} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -841\) , \( -9737\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-841{x}-9737$
16900.2-e1	16900.2-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{4} \cdot 5^{8} \cdot 13^{8} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{6} \)	$2.077762263$	$0.675991984$	8.493923856	\( -\frac{32798729601}{71402500} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -67\) , \( -441\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-67{x}-441$
16900.2-e2	16900.2-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{16} \cdot 5^{2} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	\( 2^{6} \)	$0.519440565$	$2.703967938$	8.493923856	\( \frac{33076161}{16640} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -7\) , \( -1\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-7{x}-1$
16900.2-e3	16900.2-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{8} \cdot 5^{4} \cdot 13^{4} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	\( 2^{6} \)	$1.038881131$	$1.351983969$	8.493923856	\( \frac{72043225281}{67600} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -87\) , \( -289\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-87{x}-289$
16900.2-e4	16900.2-e	$4$	$4$	\(\Q(\sqrt{-7}) \)	$2$	$[0, 1]$	16900.2	\( 2^{2} \cdot 5^{2} \cdot 13^{2} \)	\( 2^{4} \cdot 5^{2} \cdot 13^{2} \)	$2.69562$	$(a), (-a+1), (5), (13)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$4$	\( 2^{2} \)	$2.077762263$	$0.675991984$	8.493923856	\( \frac{294889639316481}{260} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -1387\) , \( -19529\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-1387{x}-19529$

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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.