Elliptic curves over Q(57)\Q(\sqrt{57}) of conductor 171.1

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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
171.1-a1	171.1-a	$2$	$5$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{10}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.2	$1$	$2^{2} \cdot 5$	$1$	$0.313808150$	0.831298097	$-\frac{9358714467168256}{22284891}$	$\bigl[0$ , $-a - 1$ , $1$ , $526841 a - 2252236$ , $404829173 a - 1730611020\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(526841a-2252236\right){x}+404829173a-1730611020$
171.1-a2	171.1-a	$2$	$5$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{26} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.1	$1$	$2^{2}$	$1$	$1.569040751$	0.831298097	$\frac{841232384}{1121931}$	$\bigl[0$ , $-a - 1$ , $1$ , $-2359 a + 10094$ , $136703 a - 584400\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2359a+10094\right){x}+136703a-584400$
171.1-b1	171.1-b	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{2}$	$0.494175486$	$3.113057546$	1.630124995	$-\frac{1413120}{19} a - \frac{13791232}{57}$	$\bigl[0$ , $a + 1$ , $1$ , $-34122 a + 145878$ , $1749942 a - 7480852\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-34122a+145878\right){x}+1749942a-7480852$
171.1-c1	171.1-c	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B	$1$	$2^{2}$	$2.365058442$	$1.080001815$	2.706567867	$-\frac{50357871050752}{19}$	$\bigl[0$ , $a + 1$ , $1$ , $-92319 a - 302343$ , $-29828922 a - 97687237\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-92319a-302343\right){x}-29828922a-97687237$
171.1-c2	171.1-c	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{6}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3Cs	$1$	$2^{2}$	$0.788352814$	$3.240005446$	2.706567867	$-\frac{89915392}{6859}$	$\bigl[0$ , $a + 1$ , $1$ , $-1119 a - 3663$ , $-44142 a - 144562\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1119a-3663\right){x}-44142a-144562$
171.1-c3	171.1-c	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B	$1$	$2^{2}$	$0.262784271$	$9.720016340$	2.706567867	$\frac{32768}{19}$	$\bigl[0$ , $a + 1$ , $1$ , $81 a + 267$ , $108 a + 353\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(81a+267\right){x}+108a+353$
171.1-d1	171.1-d	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓						$1$	$2^{2}$	$1$	$6.152988229$	3.259932801	$-\frac{1404928}{171}$	$\bigl[0$ , $a + 1$ , $1$ , $281 a - 1192$ , $-5663 a + 24214\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(281a-1192\right){x}-5663a+24214$
171.1-e1	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{8}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1$	$1.884869071$	0.998628029	$\frac{67419143}{390963}$	$\bigl[1$ , $a$ , $0$ , $1018 a + 3337$ , $104632 a + 342660\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(1018a+3337\right){x}+104632a+342660$
171.1-e2	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$- 3^{7} \cdot 19$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2$	$1$	$15.07895257$	0.998628029	$-\frac{276137246}{57} a + \frac{1180481203}{57}$	$\bigl[1$ , $a$ , $0$ , $6 a - 18$ , $-18 a + 81\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(6a-18\right){x}-18a+81$
171.1-e3	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{3}$	$1$	$15.07895257$	0.998628029	$\frac{389017}{57}$	$\bigl[1$ , $a$ , $0$ , $-182 a - 593$ , $-2528 a - 8280\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-182a-593\right){x}-2528a-8280$
171.1-e4	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{4}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{4}$	$1$	$7.539476287$	0.998628029	$\frac{30664297}{3249}$	$\bigl[1$ , $a$ , $0$ , $-782 a - 2558$ , $19744 a + 64659\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-782a-2558\right){x}+19744a+64659$
171.1-e5	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$- 3^{7} \cdot 19$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$1$	$7.539476287$	0.998628029	$\frac{276137246}{57} a + \frac{904343957}{57}$	$\bigl[1$ , $a$ , $0$ , $-9960 a + 42585$ , $-2264742 a + 9681588\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-9960a+42585\right){x}-2264742a+9681588$
171.1-e6	171.1-e	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{14} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$1$	$3.769738143$	0.998628029	$\frac{115714886617}{1539}$	$\bigl[1$ , $a$ , $0$ , $-12182 a - 39893$ , $1403704 a + 4597014\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-12182a-39893\right){x}+1403704a+4597014$
171.1-f1	171.1-f	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{2}$	$0.434454973$	$14.02540668$	6.456732522	$\frac{1413120}{19} a - \frac{18030592}{57}$	$\bigl[0$ , $a + 1$ , $1$ , $4 a - 5$ , $-5 a + 32\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-5\right){x}-5a+32$
171.1-g1	171.1-g	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{2}$	$0.434454973$	$14.02540668$	6.456732522	$-\frac{1413120}{19} a - \frac{13791232}{57}$	$\bigl[0$ , $-a - 1$ , $1$ , $-2 a - 2$ , $8 a + 29\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a-2\right){x}+8a+29$
171.1-h1	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{8}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1$	$1.884869071$	0.998628029	$\frac{67419143}{390963}$	$\bigl[1$ , $-a + 1$ , $0$ , $-1018 a + 4355$ , $-104632 a + 447292\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1018a+4355\right){x}-104632a+447292$
171.1-h2	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$- 3^{7} \cdot 19$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$1$	$7.539476287$	0.998628029	$-\frac{276137246}{57} a + \frac{1180481203}{57}$	$\bigl[1$ , $-a + 1$ , $0$ , $9960 a + 32625$ , $2264742 a + 7416846\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9960a+32625\right){x}+2264742a+7416846$
171.1-h3	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{3}$	$1$	$15.07895257$	0.998628029	$\frac{389017}{57}$	$\bigl[1$ , $-a + 1$ , $0$ , $182 a - 775$ , $2528 a - 10808\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(182a-775\right){x}+2528a-10808$
171.1-h4	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{4}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{4}$	$1$	$7.539476287$	0.998628029	$\frac{30664297}{3249}$	$\bigl[1$ , $-a + 1$ , $0$ , $782 a - 3340$ , $-19744 a + 84403\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(782a-3340\right){x}-19744a+84403$
171.1-h5	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$- 3^{7} \cdot 19$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2$	$1$	$15.07895257$	0.998628029	$\frac{276137246}{57} a + \frac{904343957}{57}$	$\bigl[1$ , $-a + 1$ , $0$ , $-6 a - 12$ , $18 a + 63\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a-12\right){x}+18a+63$
171.1-h6	171.1-h	$6$	$8$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{14} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$1$	$3.769738143$	0.998628029	$\frac{115714886617}{1539}$	$\bigl[1$ , $-a + 1$ , $0$ , $12182 a - 52075$ , $-1403704 a + 6000718\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(12182a-52075\right){x}-1403704a+6000718$
171.1-i1	171.1-i	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓						$1$	$2^{2}$	$1$	$6.152988229$	3.259932801	$-\frac{1404928}{171}$	$\bigl[0$ , $-a - 1$ , $1$ , $-279 a - 912$ , $5943 a + 19463\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-279a-912\right){x}+5943a+19463$
171.1-j1	171.1-j	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B	$1$	$2^{2}$	$2.365058442$	$1.080001815$	2.706567867	$-\frac{50357871050752}{19}$	$\bigl[0$ , $-a - 1$ , $1$ , $92321 a - 394663$ , $29736602 a - 127121496\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(92321a-394663\right){x}+29736602a-127121496$
171.1-j2	171.1-j	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{6}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3Cs	$1$	$2^{2}$	$0.788352814$	$3.240005446$	2.706567867	$-\frac{89915392}{6859}$	$\bigl[0$ , $-a - 1$ , $1$ , $1121 a - 4783$ , $43022 a - 183921\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1121a-4783\right){x}+43022a-183921$
171.1-j3	171.1-j	$3$	$9$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{6} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3$	3B	$1$	$2^{2}$	$0.262784271$	$9.720016340$	2.706567867	$\frac{32768}{19}$	$\bigl[0$ , $-a - 1$ , $1$ , $-79 a + 347$ , $-28 a + 114\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-79a+347\right){x}-28a+114$
171.1-k1	171.1-k	$1$	$1$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{8} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$							$1$	$2^{2}$	$0.494175486$	$3.113057546$	1.630124995	$\frac{1413120}{19} a - \frac{18030592}{57}$	$\bigl[0$ , $-a - 1$ , $1$ , $34124 a + 111755$ , $-1784065 a - 5842665\bigr]$	${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(34124a+111755\right){x}-1784065a-5842665$
171.1-l1	171.1-l	$2$	$5$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{10} \cdot 19^{10}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.2	$1$	$2^{2} \cdot 5$	$1$	$0.313808150$	0.831298097	$-\frac{9358714467168256}{22284891}$	$\bigl[0$ , $a + 1$ , $1$ , $-526839 a - 1725396$ , $-405356013 a - 1327507243\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-526839a-1725396\right){x}-405356013a-1327507243$
171.1-l2	171.1-l	$2$	$5$	$\Q(\sqrt{57})$	$2$	$[2, 0]$	171.1	$3^{2} \cdot 19$	$3^{26} \cdot 19^{2}$	$2.43964$	$(4a+13), (10a-43)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$5$	5B.4.1	$1$	$2^{2}$	$1$	$1.569040751$	0.831298097	$\frac{841232384}{1121931}$	$\bigl[0$ , $a + 1$ , $1$ , $2361 a + 7734$ , $-134343 a - 439963\bigr]$	${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2361a+7734\right){x}-134343a-439963$

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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.