Elliptic curves over \(\Q(\sqrt{-111}) \) of conductor 288.2

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 1000 over imaginary quadratic fields with absolute discriminant 111

Note: The completeness Only modular elliptic curves are included

Results (10 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
288.2-a1	288.2-a	$2$	$5$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{74} \cdot 3^{8} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B	$1$	\( 2^{2} \)	$7.154187014$	$0.360836430$	1.960194492	\( -\frac{136511322949}{100663296} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5140\) , \( -38996 a + 22068\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+5140\right){x}-38996a+22068$
288.2-a2	288.2-a	$2$	$5$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{34} \cdot 3^{16} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B	$1$	\( 2^{2} \)	$1.430837402$	$1.804182153$	1.960194492	\( -\frac{1295029}{7776} \)	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 100\) , \( 316 a - 108\bigr] \)	${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+100\right){x}+316a-108$
288.2-b1	288.2-b	$1$	$1$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{18} \cdot 3^{8} \cdot 5^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 2^{3} \)	$1$	$3.908053655$	2.967488299	\( \frac{293905}{128} a + \frac{3102191}{96} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -19 a - 556\) , \( -251 a - 3580\bigr] \)	${y}^2+a{x}{y}={x}^3-{x}^2+\left(-19a-556\right){x}-251a-3580$
288.2-c1	288.2-c	$2$	$2$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{28} \cdot 3^{7} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$4.630227436$	$5.753345835$	5.056980869	\( \frac{13475}{768} a + \frac{301151}{192} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( -2 a + 43\) , \( 4 a - 63\bigr] \)	${y}^2+a{x}{y}={x}^3-{x}^2+\left(-2a+43\right){x}+4a-63$
288.2-c2	288.2-c	$2$	$2$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{8} \cdot 7^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$2.315113718$	$5.753345835$	5.056980869	\( \frac{991221}{16} a + \frac{1829963}{12} \)	\( \bigl[a\) , \( -1\) , \( 0\) , \( 53 a + 743\) , \( 1336 a - 7336\bigr] \)	${y}^2+a{x}{y}={x}^3-{x}^2+\left(53a+743\right){x}+1336a-7336$
288.2-d1	288.2-d	$2$	$2$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{16} \cdot 3^{7} \cdot 5^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$1$	$5.753345835$	2.184333680	\( \frac{13475}{768} a + \frac{301151}{192} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -27 a + 88\) , \( 39 a - 204\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-27a+88\right){x}+39a-204$
288.2-d2	288.2-d	$2$	$2$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{8} \cdot 3^{8} \cdot 7^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{4} \)	$1$	$5.753345835$	2.184333680	\( \frac{991221}{16} a + \frac{1829963}{12} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 97 a - 625\) , \( -1753 a + 6532\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(97a-625\right){x}-1753a+6532$
288.2-e1	288.2-e	$1$	$1$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{30} \cdot 3^{8} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	\( 2^{2} \cdot 7 \)	$0.371645284$	$3.908053655$	7.719971234	\( \frac{293905}{128} a + \frac{3102191}{96} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( -19 a + 15\) , \( -18 a + 229\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-19a+15\right){x}-18a+229$
288.2-f1	288.2-f	$2$	$5$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{62} \cdot 3^{8} \cdot 5^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B	$1$	\( 2^{3} \cdot 5^{2} \)	$1$	$0.360836430$	6.849815668	\( -\frac{136511322949}{100663296} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 5469 a + 11620\) , \( 46311 a + 3262872\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(5469a+11620\right){x}+46311a+3262872$
288.2-f2	288.2-f	$2$	$5$	\(\Q(\sqrt{-111}) \)	$2$	$[0, 1]$	288.2	\( 2^{5} \cdot 3^{2} \)	\( 2^{22} \cdot 3^{16} \cdot 5^{12} \)	$3.87836$	$(2,a), (2,a+1), (3,a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B	$1$	\( 2^{3} \cdot 5 \)	$1$	$1.804182153$	6.849815668	\( -\frac{1295029}{7776} \)	\( \bigl[a\) , \( -1\) , \( a\) , \( 114 a + 280\) , \( -687 a - 23712\bigr] \)	${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(114a+280\right){x}-687a-23712$

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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.