LMFDB - Number field 45.45.16231657867656434784079327524611544008919694606149931176692579515382388545323258023038948936636103876569768641.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)

gp: K = bnfinit(y^45 - 4*y^44 - 182*y^43 + 580*y^42 + 15229*y^41 - 35784*y^40 - 771497*y^39 + 1192202*y^38 + 26249095*y^37 - 21282586*y^36 - 630350986*y^35 + 101314575*y^34 + 10957230656*y^33 + 4486413137*y^32 - 139417559409*y^31 - 128053969474*y^30 + 1299055132435*y^29 + 1816090021004*y^28 - 8766783201604*y^27 - 16658946907076*y^26 + 41634175647095*y^25 + 105419181053468*y^24 - 129801264303153*y^23 - 468263377533416*y^22 + 210417608171782*y^21 + 1454214352745320*y^20 + 113648989333176*y^19 - 3097196961798623*y^18 - 1450605309591570*y^17 + 4363534978783893*y^16 + 3500531720706214*y^15 - 3807203164617687*y^14 - 4497701352355764*y^13 + 1760587964112158*y^12 + 3382378542501223*y^11 - 165688879950870*y^10 - 1480388191573288*y^9 - 205996027224400*y^8 + 358811476146228*y^7 + 88160595048203*y^6 - 42989403930482*y^5 - 13014696239433*y^4 + 1879419740489*y^3 + 646702182070*y^2 + 19352782133*y - 2280393187, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)

$ x^{45} - 4 x^{44} - 182 x^{43} + 580 x^{42} + 15229 x^{41} - 35784 x^{40} - 771497 x^{39} + \cdots - 2280393187 $ x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$45$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[45, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$162\!\cdots\!641$ 16231657867656434784079327524611544008919694606149931176692579515382388545323258023038948936636103876569768641 $\medspace = 19^{40}\cdot 41^{36}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$267.24$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$19^{8/9}41^{4/5}\approx 267.2372246999024$
Ramified primes:	$19$, $41$ 19, 41	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$45$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$779=19\cdot 41$
Dirichlet character group:	$\lbrace$$\chi_{779}(256,·)$, $\chi_{779}(1,·)$, $\chi_{779}(387,·)$, $\chi_{779}(385,·)$, $\chi_{779}(264,·)$, $\chi_{779}(139,·)$, $\chi_{779}(652,·)$, $\chi_{779}(16,·)$, $\chi_{779}(529,·)$, $\chi_{779}(406,·)$, $\chi_{779}(666,·)$, $\chi_{779}(283,·)$, $\chi_{779}(543,·)$, $\chi_{779}(672,·)$, $\chi_{779}(674,·)$, $\chi_{779}(549,·)$, $\chi_{779}(42,·)$, $\chi_{779}(305,·)$, $\chi_{779}(180,·)$, $\chi_{779}(693,·)$, $\chi_{779}(182,·)$, $\chi_{779}(329,·)$, $\chi_{779}(575,·)$, $\chi_{779}(707,·)$, $\chi_{779}(324,·)$, $\chi_{779}(201,·)$, $\chi_{779}(631,·)$, $\chi_{779}(461,·)$, $\chi_{779}(590,·)$, $\chi_{779}(83,·)$, $\chi_{779}(206,·)$, $\chi_{779}(215,·)$, $\chi_{779}(346,·)$, $\chi_{779}(92,·)$, $\chi_{779}(739,·)$, $\chi_{779}(100,·)$, $\chi_{779}(657,·)$, $\chi_{779}(748,·)$, $\chi_{779}(365,·)$, $\chi_{779}(625,·)$, $\chi_{779}(370,·)$, $\chi_{779}(467,·)$, $\chi_{779}(119,·)$, $\chi_{779}(633,·)$, $\chi_{779}(510,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{3}a^{27}-\frac{1}{3}a^{21}+\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{28}-\frac{1}{3}a^{22}+\frac{1}{3}a^{20}-\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{29}-\frac{1}{3}a^{23}+\frac{1}{3}a^{21}-\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{30}-\frac{1}{3}a^{24}+\frac{1}{3}a^{22}-\frac{1}{3}a^{18}+\frac{1}{3}a^{16}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{31}-\frac{1}{3}a^{25}+\frac{1}{3}a^{23}-\frac{1}{3}a^{19}+\frac{1}{3}a^{17}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{32}-\frac{1}{3}a^{26}+\frac{1}{3}a^{24}-\frac{1}{3}a^{20}+\frac{1}{3}a^{18}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{33}+\frac{1}{3}a^{25}+\frac{1}{3}a^{21}-\frac{1}{3}a^{19}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{34}+\frac{1}{3}a^{26}+\frac{1}{3}a^{22}-\frac{1}{3}a^{20}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{35}+\frac{1}{3}a^{23}-\frac{1}{3}a^{19}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{36}+\frac{1}{9}a^{35}-\frac{1}{9}a^{34}-\frac{1}{9}a^{33}+\frac{1}{9}a^{32}-\frac{1}{9}a^{31}-\frac{1}{9}a^{30}-\frac{1}{9}a^{29}-\frac{1}{9}a^{28}+\frac{1}{9}a^{26}-\frac{2}{9}a^{23}-\frac{1}{9}a^{22}+\frac{4}{9}a^{21}+\frac{4}{9}a^{20}-\frac{2}{9}a^{19}-\frac{1}{9}a^{18}+\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{4}{9}a^{14}-\frac{4}{9}a^{13}-\frac{1}{3}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{3}a^{7}+\frac{2}{9}a^{6}+\frac{4}{9}a^{5}+\frac{2}{9}a^{4}-\frac{4}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{9}a^{37}+\frac{1}{9}a^{35}-\frac{1}{9}a^{33}+\frac{1}{9}a^{32}+\frac{1}{9}a^{28}+\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{1}{3}a^{25}+\frac{1}{9}a^{24}+\frac{4}{9}a^{23}-\frac{4}{9}a^{22}-\frac{1}{3}a^{21}+\frac{1}{9}a^{19}-\frac{2}{9}a^{18}+\frac{1}{3}a^{16}-\frac{2}{9}a^{15}-\frac{2}{9}a^{14}+\frac{1}{9}a^{13}-\frac{4}{9}a^{12}-\frac{4}{9}a^{11}+\frac{1}{9}a^{10}-\frac{4}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{38}-\frac{1}{9}a^{35}-\frac{1}{9}a^{33}-\frac{1}{9}a^{32}+\frac{1}{9}a^{31}+\frac{1}{9}a^{30}-\frac{1}{9}a^{29}-\frac{1}{9}a^{28}-\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{2}{9}a^{25}+\frac{4}{9}a^{24}+\frac{1}{9}a^{23}+\frac{1}{9}a^{22}-\frac{4}{9}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{9}a^{18}+\frac{1}{3}a^{17}-\frac{2}{9}a^{16}+\frac{4}{9}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{4}{9}a^{12}-\frac{1}{9}a^{11}-\frac{2}{9}a^{10}-\frac{1}{3}a^{9}-\frac{2}{9}a^{8}+\frac{2}{9}a^{7}-\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{39}+\frac{1}{9}a^{35}+\frac{1}{9}a^{34}+\frac{1}{9}a^{33}-\frac{1}{9}a^{32}+\frac{1}{9}a^{30}+\frac{1}{9}a^{29}+\frac{1}{9}a^{28}-\frac{1}{9}a^{27}-\frac{4}{9}a^{26}-\frac{2}{9}a^{25}+\frac{4}{9}a^{24}-\frac{4}{9}a^{23}-\frac{2}{9}a^{22}+\frac{1}{9}a^{21}+\frac{4}{9}a^{20}-\frac{1}{9}a^{19}-\frac{4}{9}a^{18}-\frac{2}{9}a^{17}-\frac{2}{9}a^{16}-\frac{2}{9}a^{14}+\frac{4}{9}a^{13}-\frac{1}{9}a^{12}-\frac{2}{9}a^{10}+\frac{1}{3}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{3}a^{4}+\frac{2}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{5139}a^{40}+\frac{3}{571}a^{39}+\frac{157}{5139}a^{38}+\frac{11}{571}a^{37}-\frac{25}{571}a^{36}-\frac{712}{5139}a^{35}-\frac{394}{5139}a^{34}+\frac{782}{5139}a^{33}+\frac{223}{5139}a^{32}-\frac{61}{571}a^{31}+\frac{13}{571}a^{30}+\frac{319}{5139}a^{29}-\frac{499}{5139}a^{28}-\frac{143}{5139}a^{27}-\frac{2380}{5139}a^{26}-\frac{742}{5139}a^{25}+\frac{637}{1713}a^{24}-\frac{239}{5139}a^{23}-\frac{279}{571}a^{22}+\frac{146}{5139}a^{21}-\frac{77}{5139}a^{20}+\frac{2506}{5139}a^{19}+\frac{643}{1713}a^{18}-\frac{14}{5139}a^{17}-\frac{1358}{5139}a^{16}+\frac{1616}{5139}a^{15}-\frac{125}{571}a^{14}-\frac{718}{1713}a^{13}+\frac{1835}{5139}a^{12}-\frac{2549}{5139}a^{11}-\frac{188}{1713}a^{10}+\frac{1948}{5139}a^{9}+\frac{449}{5139}a^{8}-\frac{856}{1713}a^{7}-\frac{1859}{5139}a^{6}-\frac{1634}{5139}a^{5}-\frac{64}{1713}a^{4}+\frac{115}{1713}a^{3}-\frac{1049}{5139}a^{2}-\frac{449}{5139}a-\frac{616}{5139}$, $\frac{1}{5139}a^{41}-\frac{1}{5139}a^{39}-\frac{143}{5139}a^{38}-\frac{43}{5139}a^{37}+\frac{224}{5139}a^{36}-\frac{584}{5139}a^{35}+\frac{236}{5139}a^{33}+\frac{94}{1713}a^{32}+\frac{94}{5139}a^{31}+\frac{5}{1713}a^{30}-\frac{547}{5139}a^{29}+\frac{256}{1713}a^{28}-\frac{232}{5139}a^{27}-\frac{2147}{5139}a^{26}-\frac{895}{5139}a^{25}+\frac{125}{5139}a^{24}-\frac{1768}{5139}a^{23}+\frac{1136}{5139}a^{22}+\frac{1120}{5139}a^{21}+\frac{17}{5139}a^{20}+\frac{1645}{5139}a^{19}+\frac{145}{1713}a^{18}-\frac{409}{5139}a^{17}+\frac{1738}{5139}a^{16}-\frac{644}{1713}a^{15}+\frac{842}{1713}a^{14}-\frac{1675}{5139}a^{13}+\frac{239}{571}a^{12}+\frac{484}{1713}a^{11}+\frac{2330}{5139}a^{10}-\frac{62}{1713}a^{9}-\frac{1558}{5139}a^{8}+\frac{11}{571}a^{7}-\frac{547}{5139}a^{6}+\frac{530}{5139}a^{5}-\frac{147}{571}a^{4}+\frac{485}{5139}a^{3}-\frac{2389}{5139}a^{2}-\frac{542}{1713}a+\frac{644}{5139}$, $\frac{1}{981549}a^{42}+\frac{3}{109061}a^{41}+\frac{74}{981549}a^{40}+\frac{46964}{981549}a^{39}-\frac{46945}{981549}a^{38}-\frac{14777}{327183}a^{37}+\frac{7426}{327183}a^{36}-\frac{142256}{981549}a^{35}-\frac{89840}{981549}a^{34}+\frac{59023}{981549}a^{33}-\frac{12385}{109061}a^{32}-\frac{31199}{981549}a^{31}-\frac{69023}{981549}a^{30}+\frac{137828}{981549}a^{29}+\frac{1922}{981549}a^{28}+\frac{49384}{981549}a^{27}+\frac{161765}{981549}a^{26}+\frac{14168}{327183}a^{25}+\frac{207742}{981549}a^{24}-\frac{265517}{981549}a^{23}+\frac{190064}{981549}a^{22}-\frac{462415}{981549}a^{21}+\frac{109387}{981549}a^{20}+\frac{156857}{981549}a^{19}-\frac{15720}{109061}a^{18}-\frac{381505}{981549}a^{17}+\frac{130432}{981549}a^{16}+\frac{488392}{981549}a^{15}+\frac{33863}{109061}a^{14}-\frac{153287}{327183}a^{13}+\frac{288514}{981549}a^{12}+\frac{232358}{981549}a^{11}+\frac{158035}{981549}a^{10}+\frac{386192}{981549}a^{9}-\frac{62537}{981549}a^{8}+\frac{99434}{327183}a^{7}-\frac{24243}{109061}a^{6}-\frac{213485}{981549}a^{5}+\frac{41180}{109061}a^{4}-\frac{354554}{981549}a^{3}+\frac{4264}{327183}a^{2}+\frac{42406}{981549}a+\frac{91669}{981549}$, $\frac{1}{635062203}a^{43}-\frac{113}{635062203}a^{42}+\frac{14057}{635062203}a^{41}-\frac{37504}{635062203}a^{40}-\frac{511703}{70562467}a^{39}+\frac{2358713}{211687401}a^{38}-\frac{32408963}{635062203}a^{37}-\frac{8891665}{635062203}a^{36}+\frac{20228819}{635062203}a^{35}+\frac{98001590}{635062203}a^{34}-\frac{33253706}{211687401}a^{33}-\frac{6412873}{635062203}a^{32}+\frac{510347}{3324933}a^{31}+\frac{87336925}{635062203}a^{30}+\frac{6406796}{211687401}a^{29}+\frac{74722783}{635062203}a^{28}+\frac{1580413}{211687401}a^{27}+\frac{253270265}{635062203}a^{26}+\frac{68465075}{635062203}a^{25}+\frac{40320527}{635062203}a^{24}-\frac{143848424}{635062203}a^{23}-\frac{164349569}{635062203}a^{22}-\frac{121405972}{635062203}a^{21}+\frac{85734034}{635062203}a^{20}-\frac{12536910}{70562467}a^{19}+\frac{59954213}{211687401}a^{18}+\frac{5269182}{70562467}a^{17}-\frac{240818641}{635062203}a^{16}-\frac{164671610}{635062203}a^{15}+\frac{73225948}{635062203}a^{14}+\frac{4978212}{70562467}a^{13}-\frac{223988698}{635062203}a^{12}-\frac{152169004}{635062203}a^{11}+\frac{267789406}{635062203}a^{10}-\frac{134177899}{635062203}a^{9}-\frac{16454174}{70562467}a^{8}-\frac{34711259}{211687401}a^{7}+\frac{122065793}{635062203}a^{6}-\frac{1416741}{70562467}a^{5}+\frac{41805829}{211687401}a^{4}+\frac{7389673}{70562467}a^{3}+\frac{199109609}{635062203}a^{2}-\frac{80971457}{211687401}a-\frac{167540222}{635062203}$, $\frac{1}{69\!\cdots\!33}a^{44}+\frac{57\!\cdots\!46}{69\!\cdots\!33}a^{43}-\frac{67\!\cdots\!70}{23\!\cdots\!11}a^{42}+\frac{21\!\cdots\!75}{23\!\cdots\!11}a^{41}-\frac{17\!\cdots\!10}{23\!\cdots\!11}a^{40}+\frac{79\!\cdots\!88}{23\!\cdots\!11}a^{39}-\frac{17\!\cdots\!78}{23\!\cdots\!11}a^{38}+\frac{90\!\cdots\!26}{77\!\cdots\!37}a^{37}+\frac{24\!\cdots\!74}{77\!\cdots\!37}a^{36}+\frac{22\!\cdots\!72}{69\!\cdots\!33}a^{35}-\frac{40\!\cdots\!67}{69\!\cdots\!33}a^{34}-\frac{27\!\cdots\!76}{69\!\cdots\!33}a^{33}-\frac{44\!\cdots\!79}{23\!\cdots\!11}a^{32}+\frac{70\!\cdots\!88}{69\!\cdots\!33}a^{31}-\frac{39\!\cdots\!99}{69\!\cdots\!33}a^{30}+\frac{79\!\cdots\!49}{69\!\cdots\!33}a^{29}+\frac{10\!\cdots\!40}{69\!\cdots\!33}a^{28}+\frac{96\!\cdots\!99}{69\!\cdots\!33}a^{27}+\frac{19\!\cdots\!03}{77\!\cdots\!37}a^{26}-\frac{26\!\cdots\!44}{69\!\cdots\!33}a^{25}-\frac{97\!\cdots\!02}{69\!\cdots\!33}a^{24}-\frac{17\!\cdots\!56}{69\!\cdots\!33}a^{23}+\frac{29\!\cdots\!90}{69\!\cdots\!33}a^{22}+\frac{28\!\cdots\!10}{69\!\cdots\!33}a^{21}-\frac{16\!\cdots\!37}{23\!\cdots\!11}a^{20}-\frac{14\!\cdots\!95}{69\!\cdots\!33}a^{19}+\frac{19\!\cdots\!19}{69\!\cdots\!33}a^{18}-\frac{99\!\cdots\!55}{23\!\cdots\!11}a^{17}+\frac{16\!\cdots\!10}{69\!\cdots\!33}a^{16}+\frac{29\!\cdots\!53}{69\!\cdots\!33}a^{15}-\frac{50\!\cdots\!42}{23\!\cdots\!11}a^{14}+\frac{77\!\cdots\!55}{77\!\cdots\!37}a^{13}-\frac{29\!\cdots\!72}{23\!\cdots\!11}a^{12}-\frac{50\!\cdots\!75}{69\!\cdots\!33}a^{11}+\frac{19\!\cdots\!18}{69\!\cdots\!33}a^{10}-\frac{42\!\cdots\!51}{23\!\cdots\!11}a^{9}-\frac{45\!\cdots\!53}{69\!\cdots\!33}a^{8}-\frac{13\!\cdots\!77}{69\!\cdots\!33}a^{7}-\frac{18\!\cdots\!80}{69\!\cdots\!33}a^{6}-\frac{24\!\cdots\!32}{69\!\cdots\!33}a^{5}+\frac{12\!\cdots\!21}{69\!\cdots\!33}a^{4}+\frac{44\!\cdots\!70}{23\!\cdots\!11}a^{3}-\frac{24\!\cdots\!40}{69\!\cdots\!33}a^{2}-\frac{10\!\cdots\!10}{23\!\cdots\!11}a+\frac{14\!\cdots\!00}{69\!\cdots\!33}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, a^20, a^21, a^22, a^23, a^24, a^25, a^26, 1/3*a^27 - 1/3*a^21 + 1/3*a^19 - 1/3*a^15 + 1/3*a^13 + 1/3*a^9 + 1/3*a^7 + 1/3*a^6 + 1/3*a^4 - 1/3*a^3 + 1/3*a^2 + 1/3*a - 1/3, 1/3*a^28 - 1/3*a^22 + 1/3*a^20 - 1/3*a^16 + 1/3*a^14 + 1/3*a^10 + 1/3*a^8 + 1/3*a^7 + 1/3*a^5 - 1/3*a^4 + 1/3*a^3 + 1/3*a^2 - 1/3*a, 1/3*a^29 - 1/3*a^23 + 1/3*a^21 - 1/3*a^17 + 1/3*a^15 + 1/3*a^11 + 1/3*a^9 + 1/3*a^8 + 1/3*a^6 - 1/3*a^5 + 1/3*a^4 + 1/3*a^3 - 1/3*a^2, 1/3*a^30 - 1/3*a^24 + 1/3*a^22 - 1/3*a^18 + 1/3*a^16 + 1/3*a^12 + 1/3*a^10 + 1/3*a^9 + 1/3*a^7 - 1/3*a^6 + 1/3*a^5 + 1/3*a^4 - 1/3*a^3, 1/3*a^31 - 1/3*a^25 + 1/3*a^23 - 1/3*a^19 + 1/3*a^17 + 1/3*a^13 + 1/3*a^11 + 1/3*a^10 + 1/3*a^8 - 1/3*a^7 + 1/3*a^6 + 1/3*a^5 - 1/3*a^4, 1/3*a^32 - 1/3*a^26 + 1/3*a^24 - 1/3*a^20 + 1/3*a^18 + 1/3*a^14 + 1/3*a^12 + 1/3*a^11 + 1/3*a^9 - 1/3*a^8 + 1/3*a^7 + 1/3*a^6 - 1/3*a^5, 1/3*a^33 + 1/3*a^25 + 1/3*a^21 - 1/3*a^19 - 1/3*a^13 + 1/3*a^12 + 1/3*a^10 + 1/3*a^8 - 1/3*a^7 + 1/3*a^4 - 1/3*a^3 + 1/3*a^2 + 1/3*a - 1/3, 1/3*a^34 + 1/3*a^26 + 1/3*a^22 - 1/3*a^20 - 1/3*a^14 + 1/3*a^13 + 1/3*a^11 + 1/3*a^9 - 1/3*a^8 + 1/3*a^5 - 1/3*a^4 + 1/3*a^3 + 1/3*a^2 - 1/3*a, 1/3*a^35 + 1/3*a^23 - 1/3*a^19 + 1/3*a^14 - 1/3*a^13 + 1/3*a^12 + 1/3*a^10 + 1/3*a^9 - 1/3*a^7 - 1/3*a^5 - 1/3*a^3 + 1/3*a^2 - 1/3*a + 1/3, 1/9*a^36 + 1/9*a^35 - 1/9*a^34 - 1/9*a^33 + 1/9*a^32 - 1/9*a^31 - 1/9*a^30 - 1/9*a^29 - 1/9*a^28 + 1/9*a^26 - 2/9*a^23 - 1/9*a^22 + 4/9*a^21 + 4/9*a^20 - 2/9*a^19 - 1/9*a^18 + 1/3*a^17 + 1/3*a^16 - 1/3*a^15 + 4/9*a^14 - 4/9*a^13 - 1/3*a^12 - 1/9*a^11 + 1/9*a^10 + 2/9*a^9 + 1/9*a^8 - 1/3*a^7 + 2/9*a^6 + 4/9*a^5 + 2/9*a^4 - 4/9*a^3 + 4/9*a^2 + 1/9*a - 1/9, 1/9*a^37 + 1/9*a^35 - 1/9*a^33 + 1/9*a^32 + 1/9*a^28 + 1/9*a^27 - 4/9*a^26 - 1/3*a^25 + 1/9*a^24 + 4/9*a^23 - 4/9*a^22 - 1/3*a^21 + 1/9*a^19 - 2/9*a^18 + 1/3*a^16 - 2/9*a^15 - 2/9*a^14 + 1/9*a^13 - 4/9*a^12 - 4/9*a^11 + 1/9*a^10 - 4/9*a^9 - 1/9*a^8 - 1/9*a^7 - 4/9*a^6 + 1/9*a^5 - 1/9*a^3 - 1/3*a^2 + 1/9*a - 2/9, 1/9*a^38 - 1/9*a^35 - 1/9*a^33 - 1/9*a^32 + 1/9*a^31 + 1/9*a^30 - 1/9*a^29 - 1/9*a^28 - 1/9*a^27 - 4/9*a^26 - 2/9*a^25 + 4/9*a^24 + 1/9*a^23 + 1/9*a^22 - 4/9*a^21 + 1/3*a^20 - 1/3*a^19 + 1/9*a^18 + 1/3*a^17 - 2/9*a^16 + 4/9*a^15 + 1/3*a^14 - 1/3*a^13 - 4/9*a^12 - 1/9*a^11 - 2/9*a^10 - 1/3*a^9 - 2/9*a^8 + 2/9*a^7 - 1/9*a^6 - 4/9*a^5 - 1/3*a^4 + 4/9*a^3 - 1/3*a^2 + 1/9, 1/9*a^39 + 1/9*a^35 + 1/9*a^34 + 1/9*a^33 - 1/9*a^32 + 1/9*a^30 + 1/9*a^29 + 1/9*a^28 - 1/9*a^27 - 4/9*a^26 - 2/9*a^25 + 4/9*a^24 - 4/9*a^23 - 2/9*a^22 + 1/9*a^21 + 4/9*a^20 - 1/9*a^19 - 4/9*a^18 - 2/9*a^17 - 2/9*a^16 - 2/9*a^14 + 4/9*a^13 - 1/9*a^12 - 2/9*a^10 + 1/3*a^8 - 1/9*a^7 - 2/9*a^6 + 1/9*a^5 + 1/3*a^4 + 2/9*a^3 + 4/9*a^2 + 2/9*a + 2/9, 1/5139*a^40 + 3/571*a^39 + 157/5139*a^38 + 11/571*a^37 - 25/571*a^36 - 712/5139*a^35 - 394/5139*a^34 + 782/5139*a^33 + 223/5139*a^32 - 61/571*a^31 + 13/571*a^30 + 319/5139*a^29 - 499/5139*a^28 - 143/5139*a^27 - 2380/5139*a^26 - 742/5139*a^25 + 637/1713*a^24 - 239/5139*a^23 - 279/571*a^22 + 146/5139*a^21 - 77/5139*a^20 + 2506/5139*a^19 + 643/1713*a^18 - 14/5139*a^17 - 1358/5139*a^16 + 1616/5139*a^15 - 125/571*a^14 - 718/1713*a^13 + 1835/5139*a^12 - 2549/5139*a^11 - 188/1713*a^10 + 1948/5139*a^9 + 449/5139*a^8 - 856/1713*a^7 - 1859/5139*a^6 - 1634/5139*a^5 - 64/1713*a^4 + 115/1713*a^3 - 1049/5139*a^2 - 449/5139*a - 616/5139, 1/5139*a^41 - 1/5139*a^39 - 143/5139*a^38 - 43/5139*a^37 + 224/5139*a^36 - 584/5139*a^35 + 236/5139*a^33 + 94/1713*a^32 + 94/5139*a^31 + 5/1713*a^30 - 547/5139*a^29 + 256/1713*a^28 - 232/5139*a^27 - 2147/5139*a^26 - 895/5139*a^25 + 125/5139*a^24 - 1768/5139*a^23 + 1136/5139*a^22 + 1120/5139*a^21 + 17/5139*a^20 + 1645/5139*a^19 + 145/1713*a^18 - 409/5139*a^17 + 1738/5139*a^16 - 644/1713*a^15 + 842/1713*a^14 - 1675/5139*a^13 + 239/571*a^12 + 484/1713*a^11 + 2330/5139*a^10 - 62/1713*a^9 - 1558/5139*a^8 + 11/571*a^7 - 547/5139*a^6 + 530/5139*a^5 - 147/571*a^4 + 485/5139*a^3 - 2389/5139*a^2 - 542/1713*a + 644/5139, 1/981549*a^42 + 3/109061*a^41 + 74/981549*a^40 + 46964/981549*a^39 - 46945/981549*a^38 - 14777/327183*a^37 + 7426/327183*a^36 - 142256/981549*a^35 - 89840/981549*a^34 + 59023/981549*a^33 - 12385/109061*a^32 - 31199/981549*a^31 - 69023/981549*a^30 + 137828/981549*a^29 + 1922/981549*a^28 + 49384/981549*a^27 + 161765/981549*a^26 + 14168/327183*a^25 + 207742/981549*a^24 - 265517/981549*a^23 + 190064/981549*a^22 - 462415/981549*a^21 + 109387/981549*a^20 + 156857/981549*a^19 - 15720/109061*a^18 - 381505/981549*a^17 + 130432/981549*a^16 + 488392/981549*a^15 + 33863/109061*a^14 - 153287/327183*a^13 + 288514/981549*a^12 + 232358/981549*a^11 + 158035/981549*a^10 + 386192/981549*a^9 - 62537/981549*a^8 + 99434/327183*a^7 - 24243/109061*a^6 - 213485/981549*a^5 + 41180/109061*a^4 - 354554/981549*a^3 + 4264/327183*a^2 + 42406/981549*a + 91669/981549, 1/635062203*a^43 - 113/635062203*a^42 + 14057/635062203*a^41 - 37504/635062203*a^40 - 511703/70562467*a^39 + 2358713/211687401*a^38 - 32408963/635062203*a^37 - 8891665/635062203*a^36 + 20228819/635062203*a^35 + 98001590/635062203*a^34 - 33253706/211687401*a^33 - 6412873/635062203*a^32 + 510347/3324933*a^31 + 87336925/635062203*a^30 + 6406796/211687401*a^29 + 74722783/635062203*a^28 + 1580413/211687401*a^27 + 253270265/635062203*a^26 + 68465075/635062203*a^25 + 40320527/635062203*a^24 - 143848424/635062203*a^23 - 164349569/635062203*a^22 - 121405972/635062203*a^21 + 85734034/635062203*a^20 - 12536910/70562467*a^19 + 59954213/211687401*a^18 + 5269182/70562467*a^17 - 240818641/635062203*a^16 - 164671610/635062203*a^15 + 73225948/635062203*a^14 + 4978212/70562467*a^13 - 223988698/635062203*a^12 - 152169004/635062203*a^11 + 267789406/635062203*a^10 - 134177899/635062203*a^9 - 16454174/70562467*a^8 - 34711259/211687401*a^7 + 122065793/635062203*a^6 - 1416741/70562467*a^5 + 41805829/211687401*a^4 + 7389673/70562467*a^3 + 199109609/635062203*a^2 - 80971457/211687401*a - 167540222/635062203, 1/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^44 + 572878460563343022814482925988102390364662358223942832797757560874432684171408145530609056111831767270113286445919305791480909947429684169844725668482080187453721098577678996116225542987599402554329226779809405349316753081649247836061046118556110246808668817596014713146/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^43 - 674726861950847624592441048004387478619171065057028804009522670365276309736543717647495152782865107977858820217925508729253491996347413923867106140377464472916508440573626815582434882717659850036129406399700077252082062197485089744246949040044978333207644839817669066611270/2318599085889122168872159534186143249941734910752763946727726763404521927443829998601803284681988724187078800757212123081172515648426168372846305217464788266617834124032528004270536007870011222771346182771894531212322682435655928329485035511450231157592802883498110181289987867211*a^42 + 219562191831840960893807996829805986408345589718997243636791280199160992022086185532100174739884734234690403790914894516376105766103213366490364700670995942463192999029668732880305441699611713358190470748829077706577405359501100797297013846846007606851721204316390767877226575/2318599085889122168872159534186143249941734910752763946727726763404521927443829998601803284681988724187078800757212123081172515648426168372846305217464788266617834124032528004270536007870011222771346182771894531212322682435655928329485035511450231157592802883498110181289987867211*a^41 - 178723389928387847453442615334350747162029202590313672946595357022494114527315786667989885356604222862302515345894355124477067825128095165797808964463794747572603778010538935363733916203841997048219626029601482109385237414860883771816326043157281597150417476002859654467143310/2318599085889122168872159534186143249941734910752763946727726763404521927443829998601803284681988724187078800757212123081172515648426168372846305217464788266617834124032528004270536007870011222771346182771894531212322682435655928329485035511450231157592802883498110181289987867211*a^40 + 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1879044525421536826210847446734303204183616042830112837041896191402909379033881945215860131785661129606207171105072458581921327689130276610632790469321861057026692024348044482028929389200737234580632890317905478053365921954082864858814776838271674185175491240210911851492309985780/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^6 - 242313224490433053634054534133998102181791699809527838950259613206243877566845612929738121109206627281514286325946796673043641316706026002339385608764614312085452402662473851024776988330621775894282295787545484641738847575391948456718012692225348719150529253942010832762011892332/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^5 + 128868567804607688186401681723222001236434558297245392714476460982551647645595438519424917294250023170534712567040473231330009840718771543200959771459609148016074494718834363693945866850970392290946676320555362911631450411524585060566932388070603492164798506574423294715387426021/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^4 + 444147053513585959959415009465562599390325630011383961819794353860334972770829262678238304165996604964530167662782545403131580758310948115297016020166660948818770719650575702328886864235065532964133021554223795997937525921764489203468795702307200393883924232011131690575460926270/2318599085889122168872159534186143249941734910752763946727726763404521927443829998601803284681988724187078800757212123081172515648426168372846305217464788266617834124032528004270536007870011222771346182771894531212322682435655928329485035511450231157592802883498110181289987867211*a^3 - 2410875365448273911850207123034612184321458987866260039817340687933595693801263621622792676345640140014499595718079901313715998755296897544634412417446025197305347037121545341681270008800716367122324126717165190090524354076488288018522566730770693098212857910739499407975589128040/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633*a^2 - 1060533462280581644960337017587369246670376760920357824107277124661848730641226597440784750291330757653910837156214792240713742792901549565597186605336974418753810902521652652653174963285769564394163675171764515019367653520462324836593021783196695404266622917609188525337283379810/2318599085889122168872159534186143249941734910752763946727726763404521927443829998601803284681988724187078800757212123081172515648426168372846305217464788266617834124032528004270536007870011222771346182771894531212322682435655928329485035511450231157592802883498110181289987867211*a + 142698449649513017997162744431095495663642580719673749410278313657811708635224597354021367051189122359626803149377025357631722441240414356903696997225957662331269696165199884573580318038171073521475766268595893400290943300871158294379083991434435634437345625437895797700053549100/6955797257667366506616478602558429749825204732258291840183180290213565782331489995805409854045966172561236402271636369243517546945278505118538915652394364799853502372097584012811608023610033668314038548315683593636968047306967784988455106534350693472778408650494330543869963601633

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$44$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	not computed	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 4*x^44 - 182*x^43 + 580*x^42 + 15229*x^41 - 35784*x^40 - 771497*x^39 + 1192202*x^38 + 26249095*x^37 - 21282586*x^36 - 630350986*x^35 + 101314575*x^34 + 10957230656*x^33 + 4486413137*x^32 - 139417559409*x^31 - 128053969474*x^30 + 1299055132435*x^29 + 1816090021004*x^28 - 8766783201604*x^27 - 16658946907076*x^26 + 41634175647095*x^25 + 105419181053468*x^24 - 129801264303153*x^23 - 468263377533416*x^22 + 210417608171782*x^21 + 1454214352745320*x^20 + 113648989333176*x^19 - 3097196961798623*x^18 - 1450605309591570*x^17 + 4363534978783893*x^16 + 3500531720706214*x^15 - 3807203164617687*x^14 - 4497701352355764*x^13 + 1760587964112158*x^12 + 3382378542501223*x^11 - 165688879950870*x^10 - 1480388191573288*x^9 - 205996027224400*x^8 + 358811476146228*x^7 + 88160595048203*x^6 - 42989403930482*x^5 - 13014696239433*x^4 + 1879419740489*x^3 + 646702182070*x^2 + 19352782133*x - 2280393187);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_{45}$ (as 45T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A cyclic group of order 45

The 45 conjugacy class representatives for $C_{45}$

Character table for $C_{45}$

Intermediate fields

3.3.361.1, 5.5.2825761.1, $\Q(\zeta_{19})^+$, 15.15.138338254038795273955595483867881.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$45$	${\href{/padicField/3.9.0.1}{9} }^{5}$	$45$	$15^{3}$	$15^{3}$	$45$	$45$	R	$45$	$45$	$15^{3}$	${\href{/padicField/37.5.0.1}{5} }^{9}$	R	$45$	$45$	$45$	$45$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$19$ 19		Deg $45$	$9$	$5$	$40$
$41$ 41		Deg $45$	$5$	$9$	$36$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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