Elliptic curves over Q(10)\Q(\sqrt{10}) of conductor 135.2

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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
135.2-a1	135.2-a	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{6} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2} \cdot 3$	$1$	$5.140502805$	2.438354577	$\frac{336226}{135} a - \frac{1062059}{135}$	$\bigl[1$ , $-a + 1$ , $a$ , $8$ , $-22 a - 69\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+8{x}-22a-69$
135.2-a2	135.2-a	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2} \cdot 3$	$1$	$5.140502805$	2.438354577	$-\frac{156415766633}{3645} a + \frac{98945821613}{729}$	$\bigl[1$ , $-a + 1$ , $a$ , $-55 a - 167$ , $-336 a - 1064\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-55a-167\right){x}-336a-1064$
135.2-b1	135.2-b	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 2^{12} \cdot 3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	$2^{2}$	$0.459288555$	$16.24433986$	2.359324574	$\frac{1026304}{3645} a + \frac{949568}{729}$	$\bigl[0$ , $a + 1$ , $0$ , $-4 a - 10$ , $-10 a - 32\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-10\right){x}-10a-32$
135.2-b2	135.2-b	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{10} \cdot 5^{6}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	$2^{2} \cdot 3$	$0.688932833$	$3.609853302$	2.359324574	$-\frac{66006784}{375} a + \frac{226222784}{375}$	$\bigl[a$ , $a$ , $1$ , $-16 a - 62$ , $-113 a - 373\bigr]$	${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-16a-62\right){x}-113a-373$
135.2-b3	135.2-b	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{6} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	$2^{2}$	$0.229644277$	$32.48867972$	2.359324574	$\frac{1217792}{135} a + \frac{4608704}{135}$	$\bigl[a$ , $a$ , $1$ , $-a - 2$ , $0\bigr]$	${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-a-2\right){x}$
135.2-b4	135.2-b	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 2^{12} \cdot 3^{11} \cdot 5^{3}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B	$1$	$2^{2} \cdot 3$	$1.377865667$	$1.804926651$	2.359324574	$\frac{74932775168}{225} a + \frac{47391625792}{45}$	$\bigl[0$ , $a + 1$ , $0$ , $-304 a - 970$ , $-5610 a - 17760\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-304a-970\right){x}-5610a-17760$
135.2-c1	135.2-c	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 3^{12} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$0.193173311$	$13.37120092$	1.633606813	$-\frac{6184497677}{3645} a + \frac{19558082533}{3645}$	$\bigl[1$ , $a$ , $0$ , $4 a + 4$ , $5 a + 41\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(4a+4\right){x}+5a+41$
135.2-c2	135.2-c	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2$	$0.386346623$	$26.74240184$	1.633606813	$\frac{211922}{135} a + \frac{64105}{27}$	$\bigl[1$ , $a$ , $0$ , $-a - 1$ , $0\bigr]$	${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-a-1\right){x}$
135.2-d1	135.2-d	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{18} \cdot 5^{4}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1.198884253$	$3.102293440$	4.704572026	$\frac{24897088}{18225}$	$\bigl[0$ , $a + 1$ , $0$ , $-48 a + 174$ , $226 a - 676\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-48a+174\right){x}+226a-676$
135.2-d2	135.2-d	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{12} \cdot 5^{8}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$2.397768506$	$6.204586881$	4.704572026	$\frac{36594368}{16875}$	$\bigl[a$ , $a$ , $1$ , $15 a - 43$ , $21 a - 67\bigr]$	${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(15a-43\right){x}+21a-67$
135.2-e1	135.2-e	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$3^{18} \cdot 5^{4}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{5} \cdot 3$	$0.088005746$	$3.102293440$	2.072073465	$\frac{24897088}{18225}$	$\bigl[a$ , $-a + 1$ , $1$ , $-15 a + 50$ , $-7 a + 24\bigr]$	${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a+50\right){x}-7a+24$
135.2-e2	135.2-e	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{12} \cdot 5^{8}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{5} \cdot 3$	$0.044002873$	$6.204586881$	2.072073465	$\frac{36594368}{16875}$	$\bigl[0$ , $-a - 1$ , $0$ , $56 a - 190$ , $242 a - 728\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(56a-190\right){x}+242a-728$
135.2-f1	135.2-f	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 2^{12} \cdot 3^{12} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3} \cdot 3$	$0.174021352$	$13.37120092$	4.414933890	$-\frac{6184497677}{3645} a + \frac{19558082533}{3645}$	$\bigl[a$ , $-a + 1$ , $a$ , $13 a + 5$ , $48 a + 178\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13a+5\right){x}+48a+178$
135.2-f2	135.2-f	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 2^{12} \cdot 3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2 \cdot 3$	$0.348042704$	$26.74240184$	4.414933890	$\frac{211922}{135} a + \frac{64105}{27}$	$\bigl[a$ , $-a + 1$ , $a$ , $-7 a - 15$ , $8 a + 30\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-15\right){x}+8a+30$
135.2-g1	135.2-g	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$2^{2} \cdot 3$	$1.661192439$	$16.24433986$	2.844466074	$\frac{1026304}{3645} a + \frac{949568}{729}$	$\bigl[a$ , $-a + 1$ , $a + 1$ , $-4 a - 1$ , $-3 a + 1\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-1\right){x}-3a+1$
135.2-g2	135.2-g	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{10} \cdot 5^{6}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$2^{2}$	$2.491788659$	$3.609853302$	2.844466074	$-\frac{66006784}{375} a + \frac{226222784}{375}$	$\bigl[0$ , $-a - 1$ , $0$ , $-68 a - 266$ , $-506 a - 1784\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-68a-266\right){x}-506a-1784$
135.2-g3	135.2-g	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{6} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$2^{2} \cdot 3$	$0.830596219$	$32.48867972$	2.844466074	$\frac{1217792}{135} a + \frac{4608704}{135}$	$\bigl[0$ , $-a - 1$ , $0$ , $-8 a - 26$ , $38 a + 120\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8a-26\right){x}+38a+120$
135.2-g4	135.2-g	$4$	$6$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$- 3^{11} \cdot 5^{3}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$2^{2}$	$4.983577319$	$1.804926651$	2.844466074	$\frac{74932775168}{225} a + \frac{47391625792}{45}$	$\bigl[a$ , $-a + 1$ , $a + 1$ , $-79 a - 241$ , $-658 a - 2080\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-79a-241\right){x}-658a-2080$
135.2-h1	135.2-h	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{6} \cdot 5^{2}$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$5.140502805$	0.812784859	$\frac{336226}{135} a - \frac{1062059}{135}$	$\bigl[a$ , $a$ , $0$ , $7 a + 22$ , $-154 a - 487\bigr]$	${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(7a+22\right){x}-154a-487$
135.2-h2	135.2-h	$2$	$2$	$\Q(\sqrt{10})$	$2$	$[2, 0]$	135.2	$3^{3} \cdot 5$	$2^{12} \cdot 3^{9} \cdot 5$	$1.92642$	$(3,a+1), (3,a+2), (5,a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$5.140502805$	0.812784859	$-\frac{156415766633}{3645} a + \frac{98945821613}{729}$	$\bigl[a$ , $a$ , $0$ , $-213 a - 678$ , $-3366 a - 10647\bigr]$	${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-213a-678\right){x}-3366a-10647$

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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.