Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 16384.1

Results (16 matches)

 

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
16384.1-a1	16384.1-a	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{16} \)	$1.75107$	$(2)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2 \)	$0.176493078$	$5.544794010$	2.260020919	\( 128 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 1\bigr] \)	${y}^2={x}^{3}+{x}^{2}+{x}+1$
16384.1-a2	16384.1-a	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{26} \)	$1.75107$	$(2)$	$2$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$0.176493078$	$2.772397005$	2.260020919	\( 10976 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -9\) , \( 7\bigr] \)	${y}^2={x}^{3}+{x}^{2}-9{x}+7$
16384.1-b1	16384.1-b	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{28} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2 \)	$1$	$2.772397005$	1.600644157	\( 128 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( 3\) , \( -5\bigr] \)	${y}^2={x}^{3}+{x}^{2}+3{x}-5$
16384.1-b2	16384.1-b	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{14} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 1 \)	$1$	$5.544794010$	1.600644157	\( 10976 \)	\( \bigl[0\) , \( 1\) , \( 0\) , \( -2\) , \( -2\bigr] \)	${y}^2={x}^{3}+{x}^{2}-2{x}-2$
16384.1-c1	16384.1-c	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{16} \)	$1.75107$	$(2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$0.435278787$	$5.276710919$	2.652162765	\( 3456 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -a\) , \( 2 a - 1\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-a{x}+2a-1$
16384.1-c2	16384.1-c	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{26} \)	$1.75107$	$(2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$0.870557574$	$2.638355459$	2.652162765	\( 23328 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -11 a\) , \( -10 a + 5\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-11a{x}-10a+5$
16384.1-d1	16384.1-d	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{28} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$1$	$2.638355459$	1.523255234	\( 3456 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -7 a\) , \( 2 a - 1\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}-7a{x}+2a-1$
16384.1-d2	16384.1-d	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{14} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 1 \)	$1$	$5.276710919$	1.523255234	\( 23328 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -2 a\) , \( -4 a + 2\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}-2a{x}-4a+2$
16384.1-e1	16384.1-e	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{16} \)	$1.75107$	$(2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$0.435278787$	$5.276710919$	2.652162765	\( 3456 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 2\) , \( 1\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+1$
16384.1-e2	16384.1-e	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{26} \)	$1.75107$	$(2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$0.870557574$	$2.638355459$	2.652162765	\( 23328 \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 12\) , \( 22 a - 5\bigr] \)	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+12\right){x}+22a-5$
16384.1-f1	16384.1-f	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{28} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 2 \)	$1$	$2.638355459$	1.523255234	\( 3456 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -7 a\) , \( -2 a + 1\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-7a{x}-2a+1$
16384.1-f2	16384.1-f	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{14} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	\( 1 \)	$1$	$5.276710919$	1.523255234	\( 23328 \)	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a\) , \( 4 a - 2\bigr] \)	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}-2a{x}+4a-2$
16384.1-g1	16384.1-g	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{28} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2 \)	$1$	$2.772397005$	1.600644157	\( 128 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( 3 a - 3\) , \( 5\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(3a-3\right){x}+5$
16384.1-g2	16384.1-g	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{14} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 1 \)	$1$	$5.544794010$	1.600644157	\( 10976 \)	\( \bigl[0\) , \( a\) , \( 0\) , \( -2 a + 2\) , \( 2\bigr] \)	${y}^2={x}^{3}+a{x}^{2}+\left(-2a+2\right){x}+2$
16384.1-h1	16384.1-h	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{16} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2 \)	$1$	$5.544794010$	3.201288314	\( 128 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -a\) , \( -1\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}-a{x}-1$
16384.1-h2	16384.1-h	$2$	$2$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	16384.1	\( 2^{14} \)	\( 2^{26} \)	$1.75107$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓		✓	$2$	2B	$1$	\( 2^{2} \)	$1$	$2.772397005$	3.201288314	\( 10976 \)	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 9 a\) , \( -7\bigr] \)	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+9a{x}-7$

 

  *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.