LMFDB - Number field 45.45.6444338306249279600681470320699578378786756892621858688031629726344906572421677992679178714752197265625.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199)

gp: K = bnfinit(y^45 - 135*y^43 - 45*y^42 + 8145*y^41 + 5076*y^40 - 290700*y^39 - 253620*y^38 + 6857235*y^37 + 7452610*y^36 - 113268978*y^35 - 144333495*y^34 + 1355213265*y^33 + 1957216455*y^32 - 11983335705*y^31 - 19266777171*y^30 + 79141746870*y^29 + 140783463765*y^28 - 391240008255*y^27 - 773615354895*y^26 + 1437267934764*y^25 + 3215872493700*y^24 - 3839971488165*y^23 - 10109340177300*y^22 + 7085017431465*y^21 + 23873005106874*y^20 - 7751347674075*y^19 - 41764000577760*y^18 + 1270029598110*y^17 + 52868417349600*y^16 + 10969424061639*y^15 - 46623586775670*y^14 - 19170786833010*y^13 + 26882974463550*y^12 + 16378966497285*y^11 - 8981328155022*y^10 - 7825331392520*y^9 + 1236586926825*y^8 + 1998991148400*y^7 + 84870189090*y^6 - 236319447555*y^5 - 31508573040*y^4 + 10090496085*y^3 + 1414058625*y^2 - 109748520*y - 10224199, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199)

$ x^{45} - 135 x^{43} - 45 x^{42} + 8145 x^{41} + 5076 x^{40} - 290700 x^{39} - 253620 x^{38} + \cdots - 10224199 $ x^45 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199

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:	$644\!\cdots\!625$ 6444338306249279600681470320699578378786756892621858688031629726344906572421677992679178714752197265625 $\medspace = 3^{110}\cdot 5^{72}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$192.60$	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:	$3^{22/9}5^{8/5}\approx 192.59650745479388$
Ramified primes:	$3$, $5$ 3, 5	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:	$675=3^{3}\cdot 5^{2}$
Dirichlet character group:	$\lbrace$$\chi_{675}(256,·)$, $\chi_{675}(1,·)$, $\chi_{675}(646,·)$, $\chi_{675}(391,·)$, $\chi_{675}(136,·)$, $\chi_{675}(526,·)$, $\chi_{675}(271,·)$, $\chi_{675}(16,·)$, $\chi_{675}(661,·)$, $\chi_{675}(406,·)$, $\chi_{675}(151,·)$, $\chi_{675}(541,·)$, $\chi_{675}(286,·)$, $\chi_{675}(31,·)$, $\chi_{675}(421,·)$, $\chi_{675}(166,·)$, $\chi_{675}(556,·)$, $\chi_{675}(301,·)$, $\chi_{675}(46,·)$, $\chi_{675}(436,·)$, $\chi_{675}(181,·)$, $\chi_{675}(571,·)$, $\chi_{675}(316,·)$, $\chi_{675}(61,·)$, $\chi_{675}(451,·)$, $\chi_{675}(196,·)$, $\chi_{675}(586,·)$, $\chi_{675}(331,·)$, $\chi_{675}(76,·)$, $\chi_{675}(466,·)$, $\chi_{675}(211,·)$, $\chi_{675}(601,·)$, $\chi_{675}(346,·)$, $\chi_{675}(91,·)$, $\chi_{675}(481,·)$, $\chi_{675}(226,·)$, $\chi_{675}(616,·)$, $\chi_{675}(361,·)$, $\chi_{675}(106,·)$, $\chi_{675}(496,·)$, $\chi_{675}(241,·)$, $\chi_{675}(631,·)$, $\chi_{675}(376,·)$, $\chi_{675}(121,·)$, $\chi_{675}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{7}a^{36}+\frac{1}{7}a^{35}-\frac{1}{7}a^{34}+\frac{3}{7}a^{33}+\frac{1}{7}a^{32}-\frac{3}{7}a^{31}+\frac{1}{7}a^{30}-\frac{1}{7}a^{29}-\frac{2}{7}a^{28}+\frac{1}{7}a^{27}-\frac{3}{7}a^{26}-\frac{3}{7}a^{25}-\frac{2}{7}a^{23}+\frac{2}{7}a^{22}+\frac{3}{7}a^{21}-\frac{2}{7}a^{20}+\frac{1}{7}a^{19}-\frac{1}{7}a^{18}+\frac{2}{7}a^{17}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{3}{7}a^{14}+\frac{3}{7}a^{12}-\frac{1}{7}a^{11}-\frac{2}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}+\frac{3}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{37}-\frac{2}{7}a^{35}-\frac{3}{7}a^{34}-\frac{2}{7}a^{33}+\frac{3}{7}a^{32}-\frac{3}{7}a^{31}-\frac{2}{7}a^{30}-\frac{1}{7}a^{29}+\frac{3}{7}a^{28}+\frac{3}{7}a^{27}+\frac{3}{7}a^{25}-\frac{2}{7}a^{24}-\frac{3}{7}a^{23}+\frac{1}{7}a^{22}+\frac{2}{7}a^{21}+\frac{3}{7}a^{20}-\frac{2}{7}a^{19}+\frac{3}{7}a^{18}+\frac{3}{7}a^{17}+\frac{1}{7}a^{16}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}+\frac{3}{7}a^{13}+\frac{3}{7}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{749}a^{38}+\frac{2}{107}a^{37}+\frac{4}{107}a^{36}+\frac{356}{749}a^{35}+\frac{178}{749}a^{34}+\frac{338}{749}a^{33}-\frac{8}{749}a^{32}+\frac{13}{749}a^{31}-\frac{13}{749}a^{30}+\frac{92}{749}a^{29}-\frac{197}{749}a^{28}-\frac{320}{749}a^{27}-\frac{87}{749}a^{26}-\frac{365}{749}a^{25}+\frac{81}{749}a^{24}-\frac{276}{749}a^{23}+\frac{279}{749}a^{22}-\frac{327}{749}a^{21}+\frac{302}{749}a^{20}-\frac{366}{749}a^{19}+\frac{358}{749}a^{18}-\frac{254}{749}a^{17}+\frac{1}{749}a^{16}+\frac{113}{749}a^{15}-\frac{353}{749}a^{14}-\frac{109}{749}a^{13}+\frac{152}{749}a^{12}+\frac{277}{749}a^{11}-\frac{44}{749}a^{10}+\frac{52}{107}a^{9}+\frac{50}{107}a^{8}+\frac{93}{749}a^{7}+\frac{38}{749}a^{6}-\frac{69}{749}a^{5}-\frac{64}{749}a^{4}+\frac{111}{749}a^{3}+\frac{351}{749}a^{2}-\frac{348}{749}a-\frac{48}{749}$, $\frac{1}{749}a^{39}+\frac{46}{749}a^{37}-\frac{36}{749}a^{36}+\frac{9}{749}a^{35}+\frac{200}{749}a^{34}+\frac{75}{749}a^{33}+\frac{18}{749}a^{32}-\frac{88}{749}a^{31}-\frac{22}{107}a^{30}-\frac{201}{749}a^{29}+\frac{12}{107}a^{28}-\frac{208}{749}a^{27}+\frac{104}{749}a^{26}-\frac{159}{749}a^{25}-\frac{340}{749}a^{24}-\frac{244}{749}a^{23}-\frac{274}{749}a^{22}+\frac{65}{749}a^{21}-\frac{207}{749}a^{20}-\frac{27}{107}a^{19}-\frac{130}{749}a^{18}-\frac{295}{749}a^{17}+\frac{313}{749}a^{16}-\frac{116}{749}a^{15}+\frac{232}{749}a^{14}+\frac{73}{749}a^{13}+\frac{289}{749}a^{12}+\frac{358}{749}a^{11}+\frac{17}{749}a^{10}+\frac{176}{749}a^{9}+\frac{222}{749}a^{8}-\frac{87}{749}a^{7}-\frac{66}{749}a^{6}+\frac{46}{749}a^{5}+\frac{151}{749}a^{4}-\frac{26}{749}a^{3}-\frac{18}{107}a^{2}-\frac{205}{749}a-\frac{184}{749}$, $\frac{1}{749}a^{40}-\frac{38}{749}a^{37}+\frac{5}{749}a^{36}+\frac{302}{749}a^{35}-\frac{88}{749}a^{34}-\frac{229}{749}a^{33}-\frac{255}{749}a^{32}+\frac{211}{749}a^{31}-\frac{352}{749}a^{30}-\frac{82}{749}a^{29}-\frac{27}{749}a^{28}+\frac{58}{749}a^{27}-\frac{9}{749}a^{26}+\frac{293}{749}a^{25}-\frac{11}{749}a^{24}-\frac{311}{749}a^{23}+\frac{178}{749}a^{22}-\frac{36}{107}a^{21}+\frac{257}{749}a^{20}+\frac{228}{749}a^{19}+\frac{51}{107}a^{18}+\frac{13}{749}a^{17}+\frac{159}{749}a^{16}-\frac{44}{749}a^{15}+\frac{22}{107}a^{14}-\frac{261}{749}a^{13}-\frac{1}{7}a^{12}+\frac{47}{107}a^{11}-\frac{261}{749}a^{10}-\frac{365}{749}a^{9}-\frac{137}{749}a^{8}+\frac{52}{107}a^{7}-\frac{97}{749}a^{6}+\frac{222}{749}a^{5}-\frac{78}{749}a^{4}+\frac{225}{749}a^{3}-\frac{194}{749}a^{2}+\frac{95}{749}a+\frac{282}{749}$, $\frac{1}{749}a^{41}+\frac{2}{749}a^{37}-\frac{25}{749}a^{36}-\frac{363}{749}a^{35}-\frac{206}{749}a^{34}-\frac{251}{749}a^{33}-\frac{93}{749}a^{32}-\frac{72}{749}a^{31}-\frac{148}{749}a^{30}+\frac{152}{749}a^{29}-\frac{37}{107}a^{28}-\frac{185}{749}a^{27}-\frac{338}{749}a^{26}-\frac{78}{749}a^{25}+\frac{92}{749}a^{24}+\frac{69}{749}a^{23}+\frac{292}{749}a^{22}-\frac{185}{749}a^{21}+\frac{148}{749}a^{20}+\frac{359}{749}a^{19}-\frac{79}{749}a^{18}+\frac{351}{749}a^{17}-\frac{6}{749}a^{16}+\frac{24}{107}a^{15}+\frac{128}{749}a^{14}+\frac{138}{749}a^{13}+\frac{327}{749}a^{12}+\frac{207}{749}a^{11}-\frac{218}{749}a^{10}+\frac{320}{749}a^{9}-\frac{139}{749}a^{8}-\frac{94}{749}a^{7}-\frac{46}{749}a^{6}+\frac{82}{749}a^{5}+\frac{21}{107}a^{4}-\frac{256}{749}a^{3}-\frac{156}{749}a^{2}+\frac{326}{749}a-\frac{5}{749}$, $\frac{1}{749}a^{42}-\frac{53}{749}a^{37}+\frac{9}{749}a^{36}+\frac{37}{107}a^{35}-\frac{286}{749}a^{34}-\frac{234}{749}a^{33}+\frac{372}{749}a^{32}+\frac{40}{749}a^{31}-\frac{143}{749}a^{30}-\frac{122}{749}a^{29}+\frac{102}{749}a^{28}-\frac{19}{749}a^{27}+\frac{310}{749}a^{26}+\frac{41}{107}a^{25}-\frac{93}{749}a^{24}-\frac{12}{749}a^{23}+\frac{113}{749}a^{22}-\frac{23}{107}a^{21}-\frac{352}{749}a^{20}+\frac{332}{749}a^{19}-\frac{44}{749}a^{18}-\frac{20}{107}a^{17}+\frac{59}{749}a^{16}+\frac{223}{749}a^{15}+\frac{309}{749}a^{14}-\frac{204}{749}a^{13}-\frac{311}{749}a^{12}+\frac{298}{749}a^{11}+\frac{43}{107}a^{10}+\frac{96}{749}a^{9}+\frac{276}{749}a^{8}-\frac{339}{749}a^{7}+\frac{6}{749}a^{6}-\frac{143}{749}a^{5}-\frac{3}{107}a^{4}+\frac{264}{749}a^{3}+\frac{373}{749}a^{2}-\frac{272}{749}a+\frac{310}{749}$, $\frac{1}{80143}a^{43}+\frac{19}{80143}a^{42}+\frac{2}{80143}a^{41}+\frac{17}{80143}a^{40}-\frac{5}{80143}a^{39}-\frac{4}{80143}a^{38}-\frac{970}{80143}a^{37}-\frac{2691}{80143}a^{36}-\frac{7905}{80143}a^{35}-\frac{15262}{80143}a^{34}-\frac{24810}{80143}a^{33}-\frac{32670}{80143}a^{32}+\frac{5886}{80143}a^{31}-\frac{16369}{80143}a^{30}+\frac{15689}{80143}a^{29}+\frac{33134}{80143}a^{28}+\frac{584}{80143}a^{27}+\frac{31916}{80143}a^{26}+\frac{588}{11449}a^{25}-\frac{19760}{80143}a^{24}-\frac{19280}{80143}a^{23}-\frac{24517}{80143}a^{22}-\frac{33615}{80143}a^{21}+\frac{6973}{80143}a^{20}+\frac{16446}{80143}a^{19}+\frac{24732}{80143}a^{18}-\frac{34156}{80143}a^{17}+\frac{35368}{80143}a^{16}-\frac{27413}{80143}a^{15}-\frac{25859}{80143}a^{14}-\frac{5661}{11449}a^{13}-\frac{16502}{80143}a^{12}-\frac{18191}{80143}a^{11}+\frac{2092}{11449}a^{10}-\frac{37869}{80143}a^{9}-\frac{16865}{80143}a^{8}+\frac{24887}{80143}a^{7}+\frac{33913}{80143}a^{6}-\frac{25523}{80143}a^{5}+\frac{4016}{11449}a^{4}+\frac{5585}{11449}a^{3}-\frac{12029}{80143}a^{2}-\frac{283}{749}a+\frac{29251}{80143}$, $\frac{1}{10\!\cdots\!51}a^{44}-\frac{61\!\cdots\!65}{10\!\cdots\!51}a^{43}+\frac{66\!\cdots\!94}{10\!\cdots\!51}a^{42}+\frac{57\!\cdots\!15}{15\!\cdots\!93}a^{41}+\frac{46\!\cdots\!44}{10\!\cdots\!51}a^{40}+\frac{14\!\cdots\!78}{10\!\cdots\!51}a^{39}-\frac{13\!\cdots\!39}{10\!\cdots\!51}a^{38}-\frac{37\!\cdots\!86}{15\!\cdots\!93}a^{37}-\frac{55\!\cdots\!18}{10\!\cdots\!51}a^{36}-\frac{13\!\cdots\!71}{10\!\cdots\!51}a^{35}+\frac{47\!\cdots\!55}{10\!\cdots\!51}a^{34}+\frac{23\!\cdots\!46}{10\!\cdots\!51}a^{33}+\frac{42\!\cdots\!68}{10\!\cdots\!51}a^{32}+\frac{36\!\cdots\!86}{10\!\cdots\!51}a^{31}+\frac{38\!\cdots\!65}{10\!\cdots\!51}a^{30}-\frac{96\!\cdots\!45}{10\!\cdots\!51}a^{29}-\frac{51\!\cdots\!60}{10\!\cdots\!51}a^{28}-\frac{38\!\cdots\!95}{10\!\cdots\!51}a^{27}+\frac{95\!\cdots\!89}{10\!\cdots\!51}a^{26}-\frac{30\!\cdots\!62}{10\!\cdots\!51}a^{25}+\frac{28\!\cdots\!35}{10\!\cdots\!51}a^{24}+\frac{13\!\cdots\!07}{10\!\cdots\!51}a^{23}+\frac{46\!\cdots\!66}{10\!\cdots\!51}a^{22}-\frac{47\!\cdots\!97}{10\!\cdots\!51}a^{21}+\frac{12\!\cdots\!16}{10\!\cdots\!51}a^{20}+\frac{19\!\cdots\!75}{10\!\cdots\!51}a^{19}-\frac{28\!\cdots\!02}{10\!\cdots\!51}a^{18}-\frac{32\!\cdots\!37}{15\!\cdots\!93}a^{17}-\frac{17\!\cdots\!69}{10\!\cdots\!51}a^{16}-\frac{21\!\cdots\!71}{10\!\cdots\!51}a^{15}-\frac{41\!\cdots\!63}{10\!\cdots\!51}a^{14}-\frac{30\!\cdots\!31}{10\!\cdots\!51}a^{13}+\frac{35\!\cdots\!20}{10\!\cdots\!51}a^{12}+\frac{22\!\cdots\!77}{10\!\cdots\!51}a^{11}+\frac{49\!\cdots\!01}{10\!\cdots\!51}a^{10}-\frac{34\!\cdots\!55}{10\!\cdots\!51}a^{9}+\frac{42\!\cdots\!55}{10\!\cdots\!51}a^{8}-\frac{34\!\cdots\!79}{10\!\cdots\!51}a^{7}-\frac{46\!\cdots\!46}{98\!\cdots\!93}a^{6}-\frac{25\!\cdots\!02}{10\!\cdots\!51}a^{5}-\frac{38\!\cdots\!22}{15\!\cdots\!93}a^{4}-\frac{70\!\cdots\!03}{15\!\cdots\!93}a^{3}-\frac{58\!\cdots\!20}{15\!\cdots\!93}a^{2}+\frac{12\!\cdots\!91}{10\!\cdots\!51}a+\frac{19\!\cdots\!91}{40\!\cdots\!07}$ 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, a^27, a^28, a^29, a^30, a^31, a^32, a^33, a^34, a^35, 1/7*a^36 + 1/7*a^35 - 1/7*a^34 + 3/7*a^33 + 1/7*a^32 - 3/7*a^31 + 1/7*a^30 - 1/7*a^29 - 2/7*a^28 + 1/7*a^27 - 3/7*a^26 - 3/7*a^25 - 2/7*a^23 + 2/7*a^22 + 3/7*a^21 - 2/7*a^20 + 1/7*a^19 - 1/7*a^18 + 2/7*a^17 - 2/7*a^16 - 1/7*a^15 - 3/7*a^14 + 3/7*a^12 - 1/7*a^11 - 2/7*a^10 - 3/7*a^9 - 1/7*a^8 - 2/7*a^7 - 1/7*a^5 + 2/7*a^4 - 2/7*a^3 + 3/7*a - 3/7, 1/7*a^37 - 2/7*a^35 - 3/7*a^34 - 2/7*a^33 + 3/7*a^32 - 3/7*a^31 - 2/7*a^30 - 1/7*a^29 + 3/7*a^28 + 3/7*a^27 + 3/7*a^25 - 2/7*a^24 - 3/7*a^23 + 1/7*a^22 + 2/7*a^21 + 3/7*a^20 - 2/7*a^19 + 3/7*a^18 + 3/7*a^17 + 1/7*a^16 - 2/7*a^15 + 3/7*a^14 + 3/7*a^13 + 3/7*a^12 - 1/7*a^11 - 1/7*a^10 + 2/7*a^9 - 1/7*a^8 + 2/7*a^7 - 1/7*a^6 + 3/7*a^5 + 3/7*a^4 + 2/7*a^3 + 3/7*a^2 + 1/7*a + 3/7, 1/749*a^38 + 2/107*a^37 + 4/107*a^36 + 356/749*a^35 + 178/749*a^34 + 338/749*a^33 - 8/749*a^32 + 13/749*a^31 - 13/749*a^30 + 92/749*a^29 - 197/749*a^28 - 320/749*a^27 - 87/749*a^26 - 365/749*a^25 + 81/749*a^24 - 276/749*a^23 + 279/749*a^22 - 327/749*a^21 + 302/749*a^20 - 366/749*a^19 + 358/749*a^18 - 254/749*a^17 + 1/749*a^16 + 113/749*a^15 - 353/749*a^14 - 109/749*a^13 + 152/749*a^12 + 277/749*a^11 - 44/749*a^10 + 52/107*a^9 + 50/107*a^8 + 93/749*a^7 + 38/749*a^6 - 69/749*a^5 - 64/749*a^4 + 111/749*a^3 + 351/749*a^2 - 348/749*a - 48/749, 1/749*a^39 + 46/749*a^37 - 36/749*a^36 + 9/749*a^35 + 200/749*a^34 + 75/749*a^33 + 18/749*a^32 - 88/749*a^31 - 22/107*a^30 - 201/749*a^29 + 12/107*a^28 - 208/749*a^27 + 104/749*a^26 - 159/749*a^25 - 340/749*a^24 - 244/749*a^23 - 274/749*a^22 + 65/749*a^21 - 207/749*a^20 - 27/107*a^19 - 130/749*a^18 - 295/749*a^17 + 313/749*a^16 - 116/749*a^15 + 232/749*a^14 + 73/749*a^13 + 289/749*a^12 + 358/749*a^11 + 17/749*a^10 + 176/749*a^9 + 222/749*a^8 - 87/749*a^7 - 66/749*a^6 + 46/749*a^5 + 151/749*a^4 - 26/749*a^3 - 18/107*a^2 - 205/749*a - 184/749, 1/749*a^40 - 38/749*a^37 + 5/749*a^36 + 302/749*a^35 - 88/749*a^34 - 229/749*a^33 - 255/749*a^32 + 211/749*a^31 - 352/749*a^30 - 82/749*a^29 - 27/749*a^28 + 58/749*a^27 - 9/749*a^26 + 293/749*a^25 - 11/749*a^24 - 311/749*a^23 + 178/749*a^22 - 36/107*a^21 + 257/749*a^20 + 228/749*a^19 + 51/107*a^18 + 13/749*a^17 + 159/749*a^16 - 44/749*a^15 + 22/107*a^14 - 261/749*a^13 - 1/7*a^12 + 47/107*a^11 - 261/749*a^10 - 365/749*a^9 - 137/749*a^8 + 52/107*a^7 - 97/749*a^6 + 222/749*a^5 - 78/749*a^4 + 225/749*a^3 - 194/749*a^2 + 95/749*a + 282/749, 1/749*a^41 + 2/749*a^37 - 25/749*a^36 - 363/749*a^35 - 206/749*a^34 - 251/749*a^33 - 93/749*a^32 - 72/749*a^31 - 148/749*a^30 + 152/749*a^29 - 37/107*a^28 - 185/749*a^27 - 338/749*a^26 - 78/749*a^25 + 92/749*a^24 + 69/749*a^23 + 292/749*a^22 - 185/749*a^21 + 148/749*a^20 + 359/749*a^19 - 79/749*a^18 + 351/749*a^17 - 6/749*a^16 + 24/107*a^15 + 128/749*a^14 + 138/749*a^13 + 327/749*a^12 + 207/749*a^11 - 218/749*a^10 + 320/749*a^9 - 139/749*a^8 - 94/749*a^7 - 46/749*a^6 + 82/749*a^5 + 21/107*a^4 - 256/749*a^3 - 156/749*a^2 + 326/749*a - 5/749, 1/749*a^42 - 53/749*a^37 + 9/749*a^36 + 37/107*a^35 - 286/749*a^34 - 234/749*a^33 + 372/749*a^32 + 40/749*a^31 - 143/749*a^30 - 122/749*a^29 + 102/749*a^28 - 19/749*a^27 + 310/749*a^26 + 41/107*a^25 - 93/749*a^24 - 12/749*a^23 + 113/749*a^22 - 23/107*a^21 - 352/749*a^20 + 332/749*a^19 - 44/749*a^18 - 20/107*a^17 + 59/749*a^16 + 223/749*a^15 + 309/749*a^14 - 204/749*a^13 - 311/749*a^12 + 298/749*a^11 + 43/107*a^10 + 96/749*a^9 + 276/749*a^8 - 339/749*a^7 + 6/749*a^6 - 143/749*a^5 - 3/107*a^4 + 264/749*a^3 + 373/749*a^2 - 272/749*a + 310/749, 1/80143*a^43 + 19/80143*a^42 + 2/80143*a^41 + 17/80143*a^40 - 5/80143*a^39 - 4/80143*a^38 - 970/80143*a^37 - 2691/80143*a^36 - 7905/80143*a^35 - 15262/80143*a^34 - 24810/80143*a^33 - 32670/80143*a^32 + 5886/80143*a^31 - 16369/80143*a^30 + 15689/80143*a^29 + 33134/80143*a^28 + 584/80143*a^27 + 31916/80143*a^26 + 588/11449*a^25 - 19760/80143*a^24 - 19280/80143*a^23 - 24517/80143*a^22 - 33615/80143*a^21 + 6973/80143*a^20 + 16446/80143*a^19 + 24732/80143*a^18 - 34156/80143*a^17 + 35368/80143*a^16 - 27413/80143*a^15 - 25859/80143*a^14 - 5661/11449*a^13 - 16502/80143*a^12 - 18191/80143*a^11 + 2092/11449*a^10 - 37869/80143*a^9 - 16865/80143*a^8 + 24887/80143*a^7 + 33913/80143*a^6 - 25523/80143*a^5 + 4016/11449*a^4 + 5585/11449*a^3 - 12029/80143*a^2 - 283/749*a + 29251/80143, 1/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^44 - 6127519752018487337784635817976604033832852926366776045836039600104692788896966704199650025375866015800156764021658196814033702041782901197724950617191604216942510394546707585742393864080355859482073123581943145049965/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^43 + 668085941137724036049948871342738958807002308693128782058308637798504114365997541457140222541882728656763388235887283674959525427129149209067283701520240816365752588172105736875350249659128730871320108718761149606633194/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^42 + 57622129396250704704080728698190284892532561824941377759820609473196560202008866866295232176188521114728801083730200573643044849241471933005658014797318533629055486343776865044645256637287500799290111716210093465304515/150787566928314522714424377854808757566183565400910806917995924667593449571859714714745002203415055244001098298914587506698413244028461325703785502226645224372224303874835942065127496294471073559159272903119119951693952993*a^41 + 465346792689620774729232213611744832839531501365536829862506024379092263989922711639600118674464418014936529456006790139131026363589264230368035101395957336843651755878429212236283743351978941629270638608985685049366744/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^40 + 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133325184853755359205693332638854928455772897973317741979479734529737871064213175444602090448031962488048348008685351178001840320276156759154197548984154353349761687292533251720136535348926241318188101612676725571370857007/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^23 + 467464944377620825500127173213841252083681385216932754104697064824948717151076937304282163642990720655675229959930687542961648362910695825178095561356886356423310860469425543518679988544646858844262039760910208902974453866/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^22 - 479371654474531790156894986197929824038869615923345572042665651416142744127755886129197266122350758544932283223085728786547890531956824653424039296977924462519226474997542068993057908255589098979725712590294094542602460397/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^21 + 127046093341732648230036505917089513347759443721562105083415475645393971294003460966927122707579634105188812264738675328193523402176240956766529902601006289465748771857251290483590256302244350845826065632175779089804999416/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^20 + 192180888830326941848229524309729432588257843959051662740862118547109357076176654715422510239906928193636473592606126047778710060444231824857878008583976761817349952887949667848805286999354075647276648267777330793422029475/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^19 - 281411060750842149492668200615400082733958566029174450283819149129693609329832773120299388129478113014242889790323801976811337079635435642434322222326178655243784049780871163564800463570598603955011510234713796468490018602/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^18 - 3276403434187065889927810925737116066062027372072562426048818598843269581445189198648117005365309130177516546316419053822089050457963688886988872749295685408367136712336423625463244969082006602253867809333723304387404537/150787566928314522714424377854808757566183565400910806917995924667593449571859714714745002203415055244001098298914587506698413244028461325703785502226645224372224303874835942065127496294471073559159272903119119951693952993*a^17 - 178204681028697163564793760624386194634321461187995923564278090751472208791766780823686908377785340046136902581044287400153573934590583694734301689049800374851432020092706095077516356245236187060074329080955226320629681469/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^16 - 214222882115289185693552352812845800909837262519380479512503670017551713166504899754447909065263077441175191459622719281724978286001413864932204549426489870008958189028277952422202763742281410109684219933030162179718887571/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^15 - 415752779085858860809039192984695257768031462872872259581995308145704530543735558112927080064664031508780409314995176442676443349230953178844494001087310835205478285975285509564703666033763950311787867587213056513456112363/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^14 - 307954942127242780154672042618530103175015279321443724190710376903316816140574258868489877650427670179329438193859444464352379891908204499622796084027990929984520793208579362845122644828502201096056607249349549434631152631/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^13 + 352835731925077209377668954923314388932969629824315591437645103393281005763696625293248920530239149613015689890313876007051244083483455176003298383982945482377809926067851672569980029873322377572126827894818973517250533820/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^12 + 223513818231548800354923522642994456290055970462494013126399111189733875047609041521953467157970485322471737846201455946424510498093765643648130250697157460960441674733667746856683237022399998151644434062027279816779265377/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^11 + 496280204750472859230329659476772113102851343796398144948883116834305233253004203093141168324519577976224908532378707830785765788865303693622780962620242290841199238766522747535310021252888213205464918127789912152059619601/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^10 - 349815569773729330208944002048449423547230592384271891772517563045933722662049139135236730721119866572170729691642956673013280904235880341563157887948376831003851223278664750749882616943915123742452581588009195240215660155/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^9 + 421008352239817353737176606902298062195812577168606436058782550001972210803720421310024728980205273569772409633134812938832915463805903635817156537368104518226535195613679489835992858889549031920005915428552738926659873255/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^8 - 3483190664487637717409545194680346102676194975756249118868446507945918441291589528529931882478930716380346563056378287229508708506008485330210914168584908539868599903380743426597825795393476899845014250761163523491246179/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^7 - 4659716821487847907523420935404764692793533207472356033153404148453039462019383749244189336473304055795963575680526086461625153753428318141103298264215588950424193927070355499005860550120122708227646126351574595829977646/9864607182226183728981034065267862644516681848657716340429639931524805112177738345824439396485097072037455028900954322868120492599992797008658864631649687575752991842278986864073761439825210419758083274035830277213623093*a^6 - 256704144448452449637882412844562084032201282776752719674484853047243471445784974362800545772460007854391172458044427339787141726238178467051853514296661724876729161955957303855390667072479506942626605854368482267280636702/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a^5 - 38914490966976445419612486907004667163276110218539704582429957283222829040044399099722155937450519817277812464347000154085321132998156529355153060636421060713606411140821932077318721152382770345537038092779123784000327022/150787566928314522714424377854808757566183565400910806917995924667593449571859714714745002203415055244001098298914587506698413244028461325703785502226645224372224303874835942065127496294471073559159272903119119951693952993*a^4 - 70189160755602891147367188930576751919462664666594218938760433080559821911383906009367454503740272277514229356509478419239658574420464835460514513626361410443246471574754747602128934412561743679643570228307702561739301403/150787566928314522714424377854808757566183565400910806917995924667593449571859714714745002203415055244001098298914587506698413244028461325703785502226645224372224303874835942065127496294471073559159272903119119951693952993*a^3 - 58223107312690091753962106277694041219456553463150560065692472389787513177353545787746773071152222371222238229458146888878358828512637862020814114778345470874888412058931585923316070834981690737334333865095389505231129320/150787566928314522714424377854808757566183565400910806917995924667593449571859714714745002203415055244001098298914587506698413244028461325703785502226645224372224303874835942065127496294471073559159272903119119951693952993*a^2 + 128814759801265488856624132877820243395581155655693826635042819396820520387577239696510030858222419113751475905373413309423133459948594847321585461497185228699278354053571759563857578435720054823391488494974178039905923591/1055512968498201659000970644983661302963284957806375648425971472673154147003018003003215015423905386708007688092402112546888892708199229279926498515586516570605570127123851594455892474061297514914114910321833839661857670951*a + 190512658583496689167625654604830254105678911933312262402370923939841894053019418756621286021168983101494313787629980388921029472036715743057609407673634807811557836947512430139485713511356838247941994738928583858856791/407062463747860261859225084837509179700456983342219687013486877236079501350951794447826847444622208526034588543155461838368257889779880169659274398606446807021045170506691706307710171253874861131552221489330443371329607

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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199)

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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199, 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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199);

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 - 135*x^43 - 45*x^42 + 8145*x^41 + 5076*x^40 - 290700*x^39 - 253620*x^38 + 6857235*x^37 + 7452610*x^36 - 113268978*x^35 - 144333495*x^34 + 1355213265*x^33 + 1957216455*x^32 - 11983335705*x^31 - 19266777171*x^30 + 79141746870*x^29 + 140783463765*x^28 - 391240008255*x^27 - 773615354895*x^26 + 1437267934764*x^25 + 3215872493700*x^24 - 3839971488165*x^23 - 10109340177300*x^22 + 7085017431465*x^21 + 23873005106874*x^20 - 7751347674075*x^19 - 41764000577760*x^18 + 1270029598110*x^17 + 52868417349600*x^16 + 10969424061639*x^15 - 46623586775670*x^14 - 19170786833010*x^13 + 26882974463550*x^12 + 16378966497285*x^11 - 8981328155022*x^10 - 7825331392520*x^9 + 1236586926825*x^8 + 1998991148400*x^7 + 84870189090*x^6 - 236319447555*x^5 - 31508573040*x^4 + 10090496085*x^3 + 1414058625*x^2 - 109748520*x - 10224199);

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

$\Q(\zeta_{9})^+$, 5.5.390625.1, $\Q(\zeta_{27})^+$, 15.15.207828545629978179931640625.1

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	$45$	R	R	${\href{/padicField/7.9.0.1}{9} }^{5}$	$45$	$45$	$15^{3}$	$15^{3}$	$45$	$45$	$45$	$15^{3}$	$45$	${\href{/padicField/43.9.0.1}{9} }^{5}$	$45$	${\href{/padicField/53.5.0.1}{5} }^{9}$	$45$

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
$3$ 3		Deg $45$	$9$	$5$	$110$
$5$ 5		Deg $45$	$5$	$9$	$72$

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