Elliptic curves over Q(5)\Q(\sqrt{5})

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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
31.1-a1	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$-31$	$0.47148$	$(5a-2)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$51.50883971$	0.359928959	$-\frac{106208}{31} a + \frac{51455}{31}$	$\bigl[1$ , $\phi + 1$ , $\phi$ , $\phi$ , $0\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\phi{x}$
31.1-a2	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$- 31^{2}$	$0.47148$	$(5a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$1.609651241$	0.359928959	$-\frac{61725871986044215714}{961} a + \frac{99874558858644938523}{961}$	$\bigl[\phi$ , $-1$ , $\phi + 1$ , $133 \phi - 141$ , $737 \phi - 764\bigr]$	${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(133\phi-141\right){x}+737\phi-764$
31.1-a3	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$31^{4}$	$0.47148$	$(5a-2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$6.438604964$	0.359928959	$-\frac{156520379364360}{923521} a + \frac{253260256463213}{923521}$	$\bigl[\phi$ , $-1$ , $\phi + 1$ , $-12 \phi - 21$ , $42 \phi + 10\bigr]$	${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-12\phi-21\right){x}+42\phi+10$
31.1-a4	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$- 31^{8}$	$0.47148$	$(5a-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$1.609651241$	0.359928959	$\frac{11889611722383394}{852891037441} a - \frac{8629385062119691}{852891037441}$	$\bigl[1$ , $\phi + 1$ , $\phi$ , $31 \phi - 75$ , $141 \phi - 303\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(31\phi-75\right){x}+141\phi-303$
31.1-a5	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$31^{2}$	$0.47148$	$(5a-2)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$25.75441985$	0.359928959	$\frac{9029272560}{961} a + \frac{5599830233}{961}$	$\bigl[1$ , $\phi + 1$ , $\phi$ , $\phi - 5$ , $3 \phi - 5\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(\phi-5\right){x}+3\phi-5$
31.1-a6	31.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.1	$31$	$-31$	$0.47148$	$(5a-2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$12.87720992$	0.359928959	$\frac{6130703730739448}{31} a + \frac{3788983280553597}{31}$	$\bigl[\phi + 1$ , $-\phi + 1$ , $\phi$ , $-18 \phi + 15$ , $171 \phi - 265\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-18\phi+15\right){x}+171\phi-265$
31.2-a1	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$- 31^{8}$	$0.47148$	$(5a-3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$1.609651241$	0.359928959	$-\frac{11889611722383394}{852891037441} a + \frac{3260226660263703}{852891037441}$	$\bigl[1$ , $-\phi - 1$ , $\phi$ , $-30 \phi - 45$ , $-111 \phi - 117\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-30\phi-45\right){x}-111\phi-117$
31.2-a2	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$-31$	$0.47148$	$(5a-3)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$12.87720992$	0.359928959	$-\frac{6130703730739448}{31} a + \frac{9919687011293045}{31}$	$\bigl[\phi$ , $1$ , $\phi + 1$ , $16 \phi - 2$ , $-172 \phi - 94\bigr]$	${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(16\phi-2\right){x}-172\phi-94$
31.2-a3	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$-31$	$0.47148$	$(5a-3)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$51.50883971$	0.359928959	$\frac{106208}{31} a - \frac{54753}{31}$	$\bigl[1$ , $-\phi - 1$ , $\phi$ , $0$ , $0\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}$
31.2-a4	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$31^{2}$	$0.47148$	$(5a-3)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$25.75441985$	0.359928959	$-\frac{9029272560}{961} a + \frac{14629102793}{961}$	$\bigl[1$ , $-\phi - 1$ , $\phi$ , $-5$ , $-3 \phi + 3\bigr]$	${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}-5{x}-3\phi+3$
31.2-a5	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$31^{4}$	$0.47148$	$(5a-3)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$6.438604964$	0.359928959	$\frac{156520379364360}{923521} a + \frac{96739877098853}{923521}$	$\bigl[\phi + 1$ , $-\phi - 1$ , $\phi$ , $10 \phi - 32$ , $-43 \phi + 53\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(10\phi-32\right){x}-43\phi+53$
31.2-a6	31.2-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	31.2	$31$	$- 31^{2}$	$0.47148$	$(5a-3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$1.609651241$	0.359928959	$\frac{61725871986044215714}{961} a + \frac{38148686872600722809}{961}$	$\bigl[\phi + 1$ , $-\phi - 1$ , $\phi$ , $-135 \phi - 7$ , $-738 \phi - 26\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-135\phi-7\right){x}-738\phi-26$
36.1-a1	36.1-a	$4$	$10$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	36.1	$2^{2} \cdot 3^{2}$	$2^{4} \cdot 3^{2}$	$0.48944$	$(2), (3)$	0	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.1[2]	$1$	$2$	$1$	$44.29962169$	0.396227861	$-\frac{24389}{12}$	$\bigl[\phi + 1$ , $\phi$ , $\phi$ , $0$ , $0\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}$
36.1-a2	36.1-a	$4$	$10$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	36.1	$2^{2} \cdot 3^{2}$	$2^{20} \cdot 3^{10}$	$0.48944$	$(2), (3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.4[2]	$1$	$2$	$1$	$1.771984867$	0.396227861	$-\frac{19465109}{248832}$	$\bigl[\phi + 1$ , $\phi$ , $\phi$ , $-5 \phi - 5$ , $-51 \phi - 37\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}+\left(-5\phi-5\right){x}-51\phi-37$
36.1-a3	36.1-a	$4$	$10$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	36.1	$2^{2} \cdot 3^{2}$	$2^{10} \cdot 3^{20}$	$0.48944$	$(2), (3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.4[2]	$1$	$2$	$1$	$1.771984867$	0.396227861	$\frac{502270291349}{1889568}$	$\bigl[\phi$ , $\phi - 1$ , $\phi$ , $165 \phi - 331$ , $1352 \phi - 2408\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(165\phi-331\right){x}+1352\phi-2408$
36.1-a4	36.1-a	$4$	$10$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	36.1	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3^{4}$	$0.48944$	$(2), (3)$	0	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.1[2]	$1$	$2$	$1$	$44.29962169$	0.396227861	$\frac{131872229}{18}$	$\bigl[\phi$ , $\phi - 1$ , $\phi$ , $10 \phi - 21$ , $-31 \phi + 51\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(10\phi-21\right){x}-31\phi+51$
41.1-a1	41.1-a	$2$	$7$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	41.1	$41$	$41$	$0.50561$	$(a+6)$	0	$\Z/7\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$7$	7B.1.1	$1$	$1$	$1$	$46.26087846$	0.422214159	$-\frac{176128}{41} a - \frac{110592}{41}$	$\bigl[0$ , $-\phi$ , $\phi$ , $0$ , $0\bigr]$	${y}^2+\phi{y}={x}^{3}-\phi{x}^{2}$
41.1-a2	41.1-a	$2$	$7$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	41.1	$41$	$41^{7}$	$0.50561$	$(a+6)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$7$	7B.1.3	$1$	$1$	$1$	$0.944099560$	0.422214159	$\frac{7215644871110656}{194754273881} a - \frac{11892928131395584}{194754273881}$	$\bigl[0$ , $-\phi$ , $\phi$ , $10 \phi - 40$ , $31 \phi - 113\bigr]$	${y}^2+\phi{y}={x}^{3}-\phi{x}^{2}+\left(10\phi-40\right){x}+31\phi-113$
41.2-a1	41.2-a	$2$	$7$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	41.2	$41$	$41^{7}$	$0.50561$	$(a-7)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$7$	7B.1.3	$1$	$1$	$1$	$0.944099560$	0.422214159	$-\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881}$	$\bigl[0$ , $\phi - 1$ , $\phi + 1$ , $-10 \phi - 30$ , $-32 \phi - 82\bigr]$	${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-10\phi-30\right){x}-32\phi-82$
41.2-a2	41.2-a	$2$	$7$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	41.2	$41$	$41$	$0.50561$	$(a-7)$	0	$\Z/7\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$7$	7B.1.1	$1$	$1$	$1$	$46.26087846$	0.422214159	$\frac{176128}{41} a - \frac{286720}{41}$	$\bigl[0$ , $\phi - 1$ , $\phi + 1$ , $0$ , $-\phi\bigr]$	${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}-\phi$
45.1-a1	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$- 3^{4} \cdot 5$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2$	2B	$16$	$2$	$1$	$0.122605555$	0.438646969	$-\frac{152409672113485069453847362}{45} a + \frac{246604029693845863366701161}{45}$	$\bigl[\phi$ , $\phi + 1$ , $1$ , $-4364 \phi - 7739$ , $-255406 \phi - 296465\bigr]$	${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(-4364\phi-7739\right){x}-255406\phi-296465$
45.1-a2	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{32} \cdot 5^{2}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	$2^{5}$	$1$	$0.490422220$	0.438646969	$-\frac{147281603041}{215233605}$	$\bigl[1$ , $1$ , $1$ , $-110$ , $-880\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$
45.1-a3	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{2} \cdot 5^{2}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	$2$	$1$	$31.38702211$	0.438646969	$-\frac{1}{15}$	$\bigl[1$ , $1$ , $1$ , $0$ , $0\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$
45.1-a4	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{4} \cdot 5^{16}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	$2^{5}$	$1$	$1.961688882$	0.438646969	$\frac{4733169839}{3515625}$	$\bigl[1$ , $1$ , $1$ , $35$ , $-28\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$
45.1-a5	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{8} \cdot 5^{8}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z\oplus\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	$2^{5}$	$1$	$7.846755528$	0.438646969	$\frac{111284641}{50625}$	$\bigl[1$ , $1$ , $1$ , $-10$ , $-10\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$
45.1-a6	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{4} \cdot 5^{4}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z\oplus\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	$2^{3}$	$1$	$31.38702211$	0.438646969	$\frac{13997521}{225}$	$\bigl[1$ , $1$ , $1$ , $-5$ , $2\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$
45.1-a7	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{16} \cdot 5^{4}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$1$	$2^{5}$	$1$	$1.961688882$	0.438646969	$\frac{272223782641}{164025}$	$\bigl[1$ , $1$ , $1$ , $-135$ , $-660\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$
45.1-a8	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{2} \cdot 5^{2}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2B	$1$	$2$	$1$	$31.38702211$	0.438646969	$\frac{56667352321}{15}$	$\bigl[1$ , $1$ , $1$ , $-80$ , $242\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$
45.1-a9	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$3^{8} \cdot 5^{2}$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$2$	2Cs	$4$	$2^{3}$	$1$	$0.490422220$	0.438646969	$\frac{1114544804970241}{405}$	$\bigl[1$ , $1$ , $1$ , $-2160$ , $-39540\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$
45.1-a10	45.1-a	$10$	$32$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	45.1	$3^{2} \cdot 5$	$- 3^{4} \cdot 5$	$0.51752$	$(-2a+1), (3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2$	2B	$16$	$2$	$1$	$0.122605555$	0.438646969	$\frac{152409672113485069453847362}{45} a + \frac{94194357580360793912853799}{45}$	$\bigl[\phi + 1$ , $\phi - 1$ , $\phi + 1$ , $4364 \phi - 12105$ , $243301 \phi - 535402\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(4364\phi-12105\right){x}+243301\phi-535402$
49.1-a1	49.1-a	$2$	$5$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	49.1	$7^{2}$	$7^{10}$	$0.52866$	$(7)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$5$	5B.1.4[2]	$1$	$1$	$1$	$1.045448192$	0.467538645	$-\frac{2887553024}{16807}$	$\bigl[0$ , $-\phi + 1$ , $1$ , $-30 \phi - 29$ , $-102 \phi - 84\bigr]$	${y}^2+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-30\phi-29\right){x}-102\phi-84$
49.1-a2	49.1-a	$2$	$5$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	49.1	$7^{2}$	$7^{2}$	$0.52866$	$(7)$	0	$\Z/5\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$5$	5B.1.1[2]	$1$	$1$	$1$	$26.13620482$	0.467538645	$\frac{4096}{7}$	$\bigl[0$ , $\phi$ , $1$ , $1$ , $0\bigr]$	${y}^2+{y}={x}^{3}+\phi{x}^{2}+{x}$
55.1-a1	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$- 5 \cdot 11^{4}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$19.86707574$	0.493601465	$-\frac{626283905886387}{73205} a + \frac{1013348626965991}{73205}$	$\bigl[1$ , $-\phi + 1$ , $1$ , $9 \phi - 25$ , $-6 \phi + 44\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(9\phi-25\right){x}-6\phi+44$
55.1-a2	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$- 5^{3} \cdot 11^{12}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.207452860$	0.493601465	$-\frac{114278307303626907}{78460709418025} a + \frac{203603378036088236}{78460709418025}$	$\bigl[1$ , $-\phi + 1$ , $1$ , $54 \phi$ , $-374 \phi - 198\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+54\phi{x}-374\phi-198$
55.1-a3	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$- 5 \cdot 11$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$1$	$1$	$39.73415148$	0.493601465	$\frac{45227}{55} a + \frac{26979}{55}$	$\bigl[\phi + 1$ , $0$ , $\phi + 1$ , $-\phi - 1$ , $-\phi\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi-1\right){x}-\phi$
55.1-a4	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$5^{6} \cdot 11^{6}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2Cs, 3B.1.2	$1$	$2^{2}$	$1$	$4.414905721$	0.493601465	$-\frac{1485675267531}{221445125} a + \frac{4152064659709}{221445125}$	$\bigl[1$ , $-\phi + 1$ , $1$ , $-21 \phi - 25$ , $-54 \phi - 58\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-21\phi-25\right){x}-54\phi-58$
55.1-a5	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$5^{12} \cdot 11^{3}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.207452860$	0.493601465	$-\frac{4560282420936767}{20796875} a + \frac{7378860561741612}{20796875}$	$\bigl[1$ , $-\phi + 1$ , $1$ , $-16 \phi - 210$ , $1110 \phi - 534\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-16\phi-210\right){x}+1110\phi-534$
55.1-a6	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$5^{2} \cdot 11^{2}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/2\Z\oplus\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2Cs, 3B.1.1	$1$	$2^{2}$	$1$	$39.73415148$	0.493601465	$\frac{132583563}{605} a + \frac{166070482}{605}$	$\bigl[\phi + 1$ , $0$ , $\phi + 1$ , $4 \phi - 11$ , $-9 \phi + 13\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(4\phi-11\right){x}-9\phi+13$
55.1-a7	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$- 5^{3} \cdot 11^{3}$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$1$	$1$	$4.414905721$	0.493601465	$\frac{754904381777}{33275} a + \frac{466557150454}{33275}$	$\bigl[\phi + 1$ , $0$ , $\phi + 1$ , $-6 \phi - 1$ , $\phi - 17\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-1\right){x}+\phi-17$
55.1-a8	55.1-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.1	$5 \cdot 11$	$5^{4} \cdot 11$	$0.54414$	$(-2a+1), (-3a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$19.86707574$	0.493601465	$\frac{48555143354501}{275} a + \frac{30008729421823}{275}$	$\bigl[\phi + 1$ , $0$ , $\phi + 1$ , $-6 \phi - 26$ , $28 \phi + 8\bigr]$	${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-6\phi-26\right){x}+28\phi+8$
55.2-a1	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$- 5 \cdot 11$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$1$	$1$	$39.73415148$	0.493601465	$-\frac{45227}{55} a + \frac{72206}{55}$	$\bigl[\phi$ , $-\phi + 1$ , $\phi$ , $-\phi$ , $0\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}-\phi{x}$
55.2-a2	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$- 5^{3} \cdot 11^{3}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$1$	$1$	$4.414905721$	0.493601465	$-\frac{754904381777}{33275} a + \frac{1221461532231}{33275}$	$\bigl[\phi$ , $-\phi + 1$ , $\phi$ , $4 \phi - 5$ , $-2 \phi - 15\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(4\phi-5\right){x}-2\phi-15$
55.2-a3	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$5^{4} \cdot 11$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$19.86707574$	0.493601465	$-\frac{48555143354501}{275} a + \frac{78563872776324}{275}$	$\bigl[\phi$ , $-\phi + 1$ , $\phi$ , $4 \phi - 30$ , $-29 \phi + 37\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(4\phi-30\right){x}-29\phi+37$
55.2-a4	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$- 5^{3} \cdot 11^{12}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.207452860$	0.493601465	$\frac{114278307303626907}{78460709418025} a + \frac{89325070732461329}{78460709418025}$	$\bigl[1$ , $\phi$ , $1$ , $-54 \phi + 54$ , $374 \phi - 572\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(-54\phi+54\right){x}+374\phi-572$
55.2-a5	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$5^{6} \cdot 11^{6}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2Cs, 3B.1.2	$1$	$2^{2}$	$1$	$4.414905721$	0.493601465	$\frac{1485675267531}{221445125} a + \frac{2666389392178}{221445125}$	$\bigl[1$ , $\phi$ , $1$ , $21 \phi - 46$ , $54 \phi - 112\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(21\phi-46\right){x}+54\phi-112$
55.2-a6	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$5^{2} \cdot 11^{2}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/2\Z\oplus\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2Cs, 3B.1.1	$1$	$2^{2}$	$1$	$39.73415148$	0.493601465	$-\frac{132583563}{605} a + \frac{59730809}{121}$	$\bigl[\phi$ , $-\phi + 1$ , $\phi$ , $-6 \phi - 5$ , $8 \phi + 5\bigr]$	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-6\phi-5\right){x}+8\phi+5$
55.2-a7	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$5^{12} \cdot 11^{3}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.207452860$	0.493601465	$\frac{4560282420936767}{20796875} a + \frac{563715628160969}{4159375}$	$\bigl[1$ , $\phi$ , $1$ , $16 \phi - 226$ , $-1110 \phi + 576\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(16\phi-226\right){x}-1110\phi+576$
55.2-a8	55.2-a	$8$	$12$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	55.2	$5 \cdot 11$	$- 5 \cdot 11^{4}$	$0.54414$	$(-2a+1), (-3a+2)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$19.86707574$	0.493601465	$\frac{626283905886387}{73205} a + \frac{387064721079604}{73205}$	$\bigl[1$ , $\phi$ , $1$ , $-9 \phi - 16$ , $6 \phi + 38\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(-9\phi-16\right){x}+6\phi+38$
64.1-a1	64.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	64.1	$2^{6}$	$- 2^{20}$	$0.56516$	$(2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	$2$	$1$	$2.397417474$	0.536078844	$-2711191688 a + 4386800300$	$\bigl[0$ , $\phi - 1$ , $0$ , $14 \phi - 25$ , $\phi - 59\bigr]$	${y}^2={x}^{3}+\left(\phi-1\right){x}^{2}+\left(14\phi-25\right){x}+\phi-59$
64.1-a2	64.1-a	$6$	$8$	$\Q(\sqrt{5})$	$2$	$[2, 0]$	64.1	$2^{6}$	$- 2^{16}$	$0.56516$	$(2)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2$	2B	$1$	$2^{2}$	$1$	$19.17933979$	0.536078844	$-548896 a + 889584$	$\bigl[0$ , $\phi - 1$ , $0$ , $4 \phi$ , $4\bigr]$	${y}^2={x}^{3}+\left(\phi-1\right){x}^{2}+4\phi{x}+4$

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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.