Elliptic curves over \(\Q(\sqrt{5}) \) of conductor 2420.1

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 (14 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
2420.1-a1	2420.1-a	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{12} \cdot 5^{6} \cdot 11^{4} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{2} \cdot 3 \)	$1$	$1.067456486$	1.432143159	\( -\frac{26537804607}{1331000} a - \frac{131169499753}{10648000} \)	\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( -14 \phi + 4\) , \( 150 \phi - 284\bigr] \)	${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(-14\phi+4\right){x}+150\phi-284$
2420.1-a2	2420.1-a	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{4} \cdot 5^{2} \cdot 11^{4} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3 \)	$1$	$9.607108375$	1.432143159	\( \frac{16187373}{26620} a + \frac{990263}{2662} \)	\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( \phi - 1\) , \( -6 \phi + 10\bigr] \)	${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}-6\phi+10$
2420.1-a3	2420.1-a	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{2} \cdot 5 \cdot 11^{8} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3 \)	$1$	$9.607108375$	1.432143159	\( -\frac{729724332409}{17715610} a + \frac{1219405135227}{17715610} \)	\( \bigl[1\) , \( -\phi + 1\) , \( \phi + 1\) , \( -8 \phi - 18\) , \( -17 \phi + 1\bigr] \)	${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-8\phi-18\right){x}-17\phi+1$
2420.1-a4	2420.1-a	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{6} \cdot 5^{3} \cdot 11^{8} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{2} \cdot 3 \)	$1$	$1.067456486$	1.432143159	\( \frac{81357485388030011}{88578050} a + \frac{201138073809942943}{354312200} \)	\( \bigl[\phi + 1\) , \( 0\) , \( 1\) , \( 186 \phi - 596\) , \( 2230 \phi - 5724\bigr] \)	${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(186\phi-596\right){x}+2230\phi-5724$
2420.1-b1	2420.1-b	$2$	$5$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{2} \cdot 5^{2} \cdot 11^{10} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$5$	5B.1.4[2]	$1$	\( 2 \cdot 5^{2} \)	$1$	$0.073934157$	1.653218027	\( -\frac{23178622194826561}{1610510} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( -5940\) , \( -178685\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5940{x}-178685$
2420.1-b2	2420.1-b	$2$	$5$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{10} \cdot 5^{10} \cdot 11^{2} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/5\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$5$	5B.1.1[2]	$1$	\( 2 \cdot 5^{2} \)	$1$	$1.848353945$	1.653218027	\( \frac{109902239}{1100000} \)	\( \bigl[1\) , \( 1\) , \( 1\) , \( 10\) , \( -45\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+10{x}-45$
2420.1-c1	2420.1-c	$2$	$3$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{6} \cdot 5^{2} \cdot 11^{2} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2 \cdot 3 \)	$0.165868134$	$21.36512140$	2.113109780	\( -\frac{117649}{440} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( -1\) , \( 1\bigr] \)	${y}^2+{x}{y}={x}^{3}-{x}+1$
2420.1-c2	2420.1-c	$2$	$3$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{2} \cdot 5^{6} \cdot 11^{6} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2 \)	$0.497604403$	$2.373902378$	2.113109780	\( \frac{80062991}{332750} \)	\( \bigl[1\) , \( 0\) , \( 0\) , \( 9\) , \( -25\bigr] \)	${y}^2+{x}{y}={x}^{3}+9{x}-25$
2420.1-d1	2420.1-d	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{4} \cdot 5^{2} \cdot 11^{4} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3 \)	$1$	$9.607108375$	1.432143159	\( -\frac{16187373}{26620} a + \frac{26090003}{26620} \)	\( \bigl[\phi\) , \( -\phi + 1\) , \( 1\) , \( -2 \phi + 1\) , \( 6 \phi + 4\bigr] \)	${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-2\phi+1\right){x}+6\phi+4$
2420.1-d2	2420.1-d	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{12} \cdot 5^{6} \cdot 11^{4} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{2} \cdot 3 \)	$1$	$1.067456486$	1.432143159	\( \frac{26537804607}{1331000} a - \frac{343471936609}{10648000} \)	\( \bigl[\phi\) , \( -\phi + 1\) , \( 1\) , \( 13 \phi - 9\) , \( -150 \phi - 134\bigr] \)	${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(13\phi-9\right){x}-150\phi-134$
2420.1-d3	2420.1-d	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{6} \cdot 5^{3} \cdot 11^{8} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	\( 2^{2} \cdot 3 \)	$1$	$1.067456486$	1.432143159	\( -\frac{81357485388030011}{88578050} a + \frac{526568015362062987}{354312200} \)	\( \bigl[\phi\) , \( -\phi + 1\) , \( 1\) , \( -187 \phi - 409\) , \( -2230 \phi - 3494\bigr] \)	${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-187\phi-409\right){x}-2230\phi-3494$
2420.1-d4	2420.1-d	$4$	$6$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{2} \cdot 5 \cdot 11^{8} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	\( 2^{2} \cdot 3 \)	$1$	$9.607108375$	1.432143159	\( \frac{729724332409}{17715610} a + \frac{244840401409}{8857805} \)	\( \bigl[1\) , \( \phi\) , \( \phi\) , \( 7 \phi - 25\) , \( 16 \phi - 15\bigr] \)	${y}^2+{x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}+\left(7\phi-25\right){x}+16\phi-15$
2420.1-e1	2420.1-e	$2$	$3$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{14} \cdot 5^{2} \cdot 11^{6} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2 \cdot 3^{2} \cdot 7 \)	$0.021400148$	$7.987258910$	2.140363659	\( -\frac{76711450249}{851840} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -89\) , \( 316\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-89{x}+316$
2420.1-e2	2420.1-e	$2$	$3$	\(\Q(\sqrt{5}) \)	$2$	$[2, 0]$	2420.1	\( 2^{2} \cdot 5 \cdot 11^{2} \)	\( 2^{42} \cdot 5^{6} \cdot 11^{2} \)	$1.40145$	$(-2a+1), (-3a+2), (-3a+1), (2)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2 \cdot 3 \cdot 7 \)	$0.064200444$	$0.887473212$	2.140363659	\( \frac{2882081488391}{2883584000} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( 296\) , \( 1702\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}+296{x}+1702$

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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.