Elliptic curves over $\Q$ of conductor 338

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Semistable	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Adelic level	Adelic index	Adelic genus	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
338.a1	338f2	338.a	338f	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2 \cdot 13^{13} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$7$	7.24.0.2	7B.6.3	$728$	$96$	$2$	$1.383448027$	$1$		$0$	$2352$	$1.572269$	$-1064019559329/125497034$	$1.06269$	$7.43052$	$[1, -1, 0, -35944, -2868878]$	\(y^2+xy=x^3-x^2-35944x-2868878\)	7.24.0.a.2, 56.48.0-7.a.2.6, 91.48.0.?, 104.2.0.?, 728.96.2.?	$[(3613/3, 180227/3)]$
338.a2	338f1	338.a	338f	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2^{7} \cdot 13^{7} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$7$	7.24.0.1	7B.6.1	$728$	$96$	$2$	$0.197635432$	$1$		$6$	$336$	$0.599314$	$-2146689/1664$	$0.96784$	$5.29260$	$[1, -1, 0, -454, 5812]$	\(y^2+xy=x^3-x^2-454x+5812\)	7.24.0.a.1, 56.48.0-7.a.1.6, 91.48.0.?, 104.2.0.?, 728.96.2.?	$[(23, 73)]$
338.b1	338d2	338.b	338d	$2$	$5$	\( 2 \cdot 13^{2} \)	\( - 2^{15} \cdot 13^{9} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3, 5$	3.6.0.1, 5.6.0.1	3Ns, 5B	$1560$	$576$	$17$	$1$	$1$		$0$	$1560$	$1.583616$	$-1680914269/32768$	$1.02322$	$7.61805$	$[1, 1, 0, -54421, 4945517]$	\(y^2+xy=x^3+x^2-54421x+4945517\)	3.6.0.b.1, 5.6.0.a.1, 15.72.1.a.1, 24.12.0.bx.1, 39.12.0.a.1, $\ldots$	$[]$
338.b2	338d1	338.b	338d	$2$	$5$	\( 2 \cdot 13^{2} \)	\( - 2^{3} \cdot 13^{9} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3, 5$	3.6.0.1, 5.6.0.1	3Ns, 5B	$1560$	$576$	$17$	$1$	$1$		$0$	$312$	$0.778896$	$1331/8$	$0.93577$	$5.58427$	$[1, 1, 0, 504, -13112]$	\(y^2+xy=x^3+x^2+504x-13112\)	3.6.0.b.1, 5.6.0.a.1, 15.72.1.a.2, 24.12.0.bx.1, 39.12.0.a.1, $\ldots$	$[]$
338.c1	338a2	338.c	338a	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2^{14} \cdot 13^{2} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 7$	4.8.0.2, 7.8.0.1	7B	$364$	$768$	$21$	$1.188700458$	$1$		$4$	$84$	$0.196927$	$-38575685889/16384$	$1.08547$	$5.06720$	$[1, -1, 0, -389, -2859]$	\(y^2+xy=x^3-x^2-389x-2859\)	4.8.0.b.1, 7.8.0.a.1, 28.128.5.b.1, 52.16.0-4.b.1.1, 91.48.0.?, $\ldots$	$[(26, 51)]$
338.c2	338a1	338.c	338a	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2^{2} \cdot 13^{2} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 7$	4.8.0.2, 7.8.0.1	7B	$364$	$768$	$21$	$0.169814351$	$1$		$6$	$12$	$-0.776028$	$351/4$	$1.27279$	$2.39029$	$[1, -1, 0, 1, 1]$	\(y^2+xy=x^3-x^2+x+1\)	4.8.0.b.1, 7.8.0.a.1, 28.128.5.b.2, 52.16.0-4.b.1.1, 91.48.0.?, $\ldots$	$[(0, 1)]$
338.d1	338e2	338.d	338e	$2$	$5$	\( 2 \cdot 13^{2} \)	\( - 2^{15} \cdot 13^{3} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3, 5$	3.6.0.1, 5.6.0.1	3Ns, 5B	$1560$	$576$	$17$	$0.030736550$	$1$		$16$	$120$	$0.301141$	$-1680914269/32768$	$1.02322$	$4.97516$	$[1, 1, 1, -322, 2127]$	\(y^2+xy+y=x^3+x^2-322x+2127\)	3.6.0.b.1, 5.6.0.a.1, 15.72.1.a.1, 24.12.0.bx.1, 39.12.0.a.1, $\ldots$	$[(5, 23)]$
338.d2	338e1	338.d	338e	$2$	$5$	\( 2 \cdot 13^{2} \)	\( - 2^{3} \cdot 13^{3} \)	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3, 5$	3.6.0.1, 5.6.0.1	3Ns, 5B	$1560$	$576$	$17$	$0.153682750$	$1$		$6$	$24$	$-0.503578$	$1331/8$	$0.93577$	$2.94138$	$[1, 1, 1, 3, -5]$	\(y^2+xy+y=x^3+x^2+3x-5\)	3.6.0.b.1, 5.6.0.a.1, 15.72.1.a.2, 24.12.0.bx.1, 39.12.0.a.1, $\ldots$	$[(5, 10)]$
338.e1	338b2	338.e	338b	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2^{14} \cdot 13^{8} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 7$	4.16.0.2, 7.16.0.2	7B.2.3	$364$	$768$	$21$	$1$	$1$		$0$	$1092$	$1.479403$	$-38575685889/16384$	$1.08547$	$7.71009$	$[1, -1, 1, -65773, -6478507]$	\(y^2+xy+y=x^3-x^2-65773x-6478507\)	4.16.0-4.b.1.1, 7.16.0-7.a.1.1, 28.256.5-28.b.1.2, 91.48.0.?, 364.768.21.?	$[]$
338.e2	338b1	338.e	338b	$2$	$7$	\( 2 \cdot 13^{2} \)	\( - 2^{2} \cdot 13^{8} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$2, 7$	4.16.0.2, 7.16.0.1	7B.2.1	$364$	$768$	$21$	$1$	$1$		$0$	$156$	$0.506447$	$351/4$	$1.27279$	$5.03319$	$[1, -1, 1, 137, 2643]$	\(y^2+xy+y=x^3-x^2+137x+2643\)	4.16.0-4.b.1.1, 7.16.0-7.a.1.2, 28.256.5-28.b.2.2, 91.48.0.?, 364.768.21.?	$[]$
338.f1	338c3	338.f	338c	$3$	$9$	\( 2 \cdot 13^{2} \)	\( - 2^{9} \cdot 13^{7} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	9.12.0.1	3B	$936$	$144$	$3$	$1$	$1$		$0$	$1008$	$1.336861$	$-10730978619193/6656$	$1.02193$	$7.79555$	$[1, 0, 0, -77659, -8336303]$	\(y^2+xy=x^3-77659x-8336303\)	3.4.0.a.1, 9.12.0.a.1, 24.8.0-3.a.1.7, 39.8.0-3.a.1.2, 72.24.0.?, $\ldots$	$[]$
338.f2	338c2	338.f	338c	$3$	$9$	\( 2 \cdot 13^{2} \)	\( - 2^{3} \cdot 13^{9} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	3.12.0.1	3Cs	$936$	$144$	$3$	$1$	$1$		$0$	$336$	$0.787555$	$-10218313/17576$	$0.94717$	$5.65146$	$[1, 0, 0, -764, -16264]$	\(y^2+xy=x^3-764x-16264\)	3.12.0.a.1, 24.24.0-3.a.1.4, 39.24.0-3.a.1.1, 104.2.0.?, 117.72.0.?, $\ldots$	$[]$
338.f3	338c1	338.f	338c	$3$	$9$	\( 2 \cdot 13^{2} \)	\( - 2 \cdot 13^{7} \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$			$3$	9.12.0.1	3B	$936$	$144$	$3$	$1$	$1$		$0$	$112$	$0.238249$	$12167/26$	$0.84415$	$4.42839$	$[1, 0, 0, 81, 467]$	\(y^2+xy=x^3+81x+467\)	3.4.0.a.1, 9.12.0.a.1, 24.8.0-3.a.1.8, 39.8.0-3.a.1.1, 72.24.0.?, $\ldots$	$[]$

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