Properties

Label 45.45.105...761.1
Degree $45$
Signature $[45, 0]$
Discriminant $1.055\times 10^{112}$
Root discriminant \(308.60\)
Ramified primes $11,73$
Class number not computed
Class group not computed
Galois group $C_{45}$ (as 45T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 4*x^44 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321)
 
gp: K = bnfinit(y^45 - 4*y^44 - 194*y^43 + 836*y^42 + 16500*y^41 - 76882*y^40 - 811248*y^39 + 4130184*y^38 + 25559474*y^37 - 145049280*y^36 - 538597921*y^35 + 3530594867*y^34 + 7615564821*y^33 - 61610572814*y^32 - 68317895227*y^31 + 786429242050*y^30 + 287086043611*y^29 - 7425134094508*y^28 + 1391788361320*y^27 + 52087220030130*y^26 - 32183862662002*y^25 - 271116028008606*y^24 + 264064430436474*y^23 + 1039304723755570*y^22 - 1345885450620938*y^21 - 2889398096911074*y^20 + 4691240971935913*y^19 + 5661912712400497*y^18 - 11481658937434788*y^17 - 7388527635358822*y^16 + 19748315799639590*y^15 + 5543796759733632*y^14 - 23515522924776395*y^13 - 895060410571712*y^12 + 18817979236586386*y^11 - 2391413888018270*y^10 - 9652129249107031*y^9 + 2328285637187904*y^8 + 2951322314183680*y^7 - 960094366581828*y^6 - 477351520888160*y^5 + 190413802657905*y^4 + 31093339218140*y^3 - 15654659537914*y^2 - 188842256201*y + 306884845321, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 4*x^44 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 4*x^44 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321)
 

\( x^{45} - 4 x^{44} - 194 x^{43} + 836 x^{42} + 16500 x^{41} - 76882 x^{40} - 811248 x^{39} + 4130184 x^{38} + 25559474 x^{37} - 145049280 x^{36} + \cdots + 306884845321 \) Copy content Toggle raw display

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:   \(105\!\cdots\!761\) \(\medspace = 11^{36}\cdot 73^{40}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(308.60\)
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:  $11^{4/5}73^{8/9}\approx 308.6037043871589$
Ramified primes:   \(11\), \(73\) Copy content Toggle raw display
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:  \(803=11\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{803}(256,·)$, $\chi_{803}(1,·)$, $\chi_{803}(515,·)$, $\chi_{803}(4,·)$, $\chi_{803}(137,·)$, $\chi_{803}(397,·)$, $\chi_{803}(621,·)$, $\chi_{803}(16,·)$, $\chi_{803}(785,·)$, $\chi_{803}(658,·)$, $\chi_{803}(147,·)$, $\chi_{803}(148,·)$, $\chi_{803}(661,·)$, $\chi_{803}(665,·)$, $\chi_{803}(543,·)$, $\chi_{803}(548,·)$, $\chi_{803}(37,·)$, $\chi_{803}(300,·)$, $\chi_{803}(566,·)$, $\chi_{803}(575,·)$, $\chi_{803}(64,·)$, $\chi_{803}(324,·)$, $\chi_{803}(694,·)$, $\chi_{803}(454,·)$, $\chi_{803}(201,·)$, $\chi_{803}(586,·)$, $\chi_{803}(75,·)$, $\chi_{803}(588,·)$, $\chi_{803}(592,·)$, $\chi_{803}(81,·)$, $\chi_{803}(210,·)$, $\chi_{803}(89,·)$, $\chi_{803}(731,·)$, $\chi_{803}(221,·)$, $\chi_{803}(223,·)$, $\chi_{803}(738,·)$, $\chi_{803}(356,·)$, $\chi_{803}(746,·)$, $\chi_{803}(235,·)$, $\chi_{803}(493,·)$, $\chi_{803}(366,·)$, $\chi_{803}(367,·)$, $\chi_{803}(762,·)$, $\chi_{803}(251,·)$, $\chi_{803}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{3}a^{30}-\frac{1}{3}a^{29}-\frac{1}{3}a^{27}-\frac{1}{3}a^{25}-\frac{1}{3}a^{24}+\frac{1}{3}a^{22}+\frac{1}{3}a^{21}+\frac{1}{3}a^{19}-\frac{1}{3}a^{18}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{31}-\frac{1}{3}a^{29}-\frac{1}{3}a^{28}-\frac{1}{3}a^{27}-\frac{1}{3}a^{26}+\frac{1}{3}a^{25}-\frac{1}{3}a^{24}+\frac{1}{3}a^{23}-\frac{1}{3}a^{22}+\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{18}+\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{32}+\frac{1}{3}a^{29}-\frac{1}{3}a^{28}+\frac{1}{3}a^{27}+\frac{1}{3}a^{26}+\frac{1}{3}a^{25}-\frac{1}{3}a^{23}-\frac{1}{3}a^{22}-\frac{1}{3}a^{21}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{591}a^{33}+\frac{17}{591}a^{32}+\frac{64}{591}a^{31}+\frac{71}{591}a^{30}-\frac{220}{591}a^{29}+\frac{229}{591}a^{28}+\frac{55}{591}a^{27}-\frac{82}{591}a^{26}-\frac{286}{591}a^{25}+\frac{33}{197}a^{24}-\frac{140}{591}a^{23}+\frac{68}{197}a^{22}-\frac{6}{197}a^{21}-\frac{101}{591}a^{20}+\frac{151}{591}a^{19}-\frac{29}{591}a^{18}-\frac{101}{591}a^{17}+\frac{154}{591}a^{16}-\frac{32}{197}a^{15}-\frac{25}{197}a^{14}-\frac{187}{591}a^{13}-\frac{179}{591}a^{12}+\frac{112}{591}a^{11}-\frac{36}{197}a^{10}-\frac{182}{591}a^{9}-\frac{193}{591}a^{8}-\frac{61}{197}a^{7}+\frac{64}{197}a^{6}+\frac{90}{197}a^{5}+\frac{37}{197}a^{4}+\frac{113}{591}a^{3}-\frac{224}{591}a^{2}-\frac{104}{591}a+\frac{1}{3}$, $\frac{1}{591}a^{34}-\frac{28}{591}a^{32}-\frac{32}{591}a^{31}-\frac{16}{197}a^{30}+\frac{29}{591}a^{29}-\frac{292}{591}a^{28}-\frac{229}{591}a^{27}-\frac{271}{591}a^{26}+\frac{12}{197}a^{25}-\frac{50}{591}a^{24}-\frac{58}{197}a^{23}+\frac{257}{591}a^{22}+\frac{8}{591}a^{21}-\frac{34}{197}a^{20}-\frac{35}{591}a^{19}-\frac{199}{591}a^{18}-\frac{33}{197}a^{17}-\frac{51}{197}a^{16}-\frac{72}{197}a^{15}+\frac{103}{591}a^{14}+\frac{15}{197}a^{13}-\frac{194}{591}a^{12}-\frac{239}{591}a^{11}+\frac{275}{591}a^{10}-\frac{18}{197}a^{9}+\frac{143}{591}a^{8}-\frac{81}{197}a^{7}+\frac{158}{591}a^{6}+\frac{52}{591}a^{5}-\frac{66}{197}a^{4}+\frac{73}{197}a^{3}-\frac{236}{591}a^{2}-\frac{5}{591}a$, $\frac{1}{591}a^{35}+\frac{50}{591}a^{32}-\frac{29}{591}a^{31}+\frac{47}{591}a^{30}-\frac{148}{591}a^{29}+\frac{76}{591}a^{28}-\frac{110}{591}a^{27}-\frac{290}{591}a^{26}+\frac{19}{591}a^{25}-\frac{160}{591}a^{24}+\frac{277}{591}a^{23}+\frac{7}{591}a^{22}+\frac{182}{591}a^{21}+\frac{92}{591}a^{20}+\frac{286}{591}a^{19}-\frac{41}{197}a^{18}-\frac{26}{591}a^{17}-\frac{238}{591}a^{16}-\frac{8}{197}a^{15}+\frac{112}{591}a^{14}-\frac{37}{197}a^{13}-\frac{43}{197}a^{12}+\frac{62}{591}a^{11}-\frac{41}{197}a^{10}-\frac{28}{591}a^{9}+\frac{263}{591}a^{8}+\frac{52}{197}a^{7}-\frac{95}{197}a^{6}-\frac{124}{591}a^{5}-\frac{73}{197}a^{4}+\frac{170}{591}a^{3}-\frac{170}{591}a^{2}-\frac{154}{591}a+\frac{1}{3}$, $\frac{1}{591}a^{36}-\frac{91}{591}a^{32}-\frac{1}{591}a^{31}+\frac{15}{197}a^{30}+\frac{241}{591}a^{29}-\frac{134}{591}a^{28}-\frac{94}{197}a^{27}-\frac{6}{197}a^{26}+\frac{51}{197}a^{25}+\frac{84}{197}a^{24}-\frac{85}{591}a^{23}-\frac{56}{197}a^{22}+\frac{7}{591}a^{21}+\frac{214}{591}a^{20}+\frac{69}{197}a^{19}-\frac{152}{591}a^{18}+\frac{281}{591}a^{17}+\frac{52}{197}a^{16}-\frac{70}{197}a^{15}+\frac{31}{197}a^{14}-\frac{38}{591}a^{13}-\frac{247}{591}a^{12}-\frac{69}{197}a^{11}+\frac{250}{591}a^{10}-\frac{31}{197}a^{9}+\frac{51}{197}a^{8}+\frac{42}{197}a^{6}+\frac{71}{591}a^{5}+\frac{136}{591}a^{4}+\frac{30}{197}a^{3}-\frac{61}{197}a^{2}+\frac{275}{591}a+\frac{1}{3}$, $\frac{1}{591}a^{37}-\frac{10}{197}a^{32}-\frac{41}{591}a^{31}+\frac{4}{591}a^{30}-\frac{257}{591}a^{29}+\frac{266}{591}a^{28}+\frac{62}{591}a^{27}-\frac{20}{591}a^{26}+\frac{11}{197}a^{25}+\frac{256}{591}a^{24}-\frac{103}{591}a^{23}-\frac{48}{197}a^{22}-\frac{15}{197}a^{21}-\frac{119}{591}a^{20}-\frac{67}{197}a^{19}+\frac{203}{591}a^{18}-\frac{170}{591}a^{17}+\frac{14}{591}a^{16}-\frac{172}{591}a^{15}+\frac{32}{591}a^{14}-\frac{125}{591}a^{13}-\frac{145}{591}a^{12}+\frac{1}{591}a^{11}+\frac{42}{197}a^{10}-\frac{85}{197}a^{9}+\frac{167}{591}a^{8}-\frac{176}{591}a^{7}+\frac{10}{591}a^{6}+\frac{27}{197}a^{5}+\frac{48}{197}a^{4}+\frac{250}{591}a^{3}+\frac{182}{591}a^{2}-\frac{8}{591}a+\frac{1}{3}$, $\frac{1}{591}a^{38}+\frac{25}{197}a^{32}-\frac{46}{591}a^{31}-\frac{97}{591}a^{30}+\frac{167}{591}a^{29}-\frac{160}{591}a^{28}-\frac{143}{591}a^{27}-\frac{260}{591}a^{26}+\frac{49}{197}a^{25}-\frac{95}{197}a^{24}-\frac{10}{591}a^{23}-\frac{32}{591}a^{22}-\frac{68}{591}a^{21}+\frac{118}{591}a^{20}-\frac{64}{197}a^{19}-\frac{55}{591}a^{18}-\frac{86}{197}a^{17}+\frac{38}{197}a^{16}-\frac{30}{197}a^{15}+\frac{62}{197}a^{14}-\frac{239}{591}a^{13}-\frac{247}{591}a^{12}-\frac{257}{591}a^{11}-\frac{146}{591}a^{10}+\frac{223}{591}a^{9}-\frac{253}{591}a^{8}+\frac{233}{591}a^{7}-\frac{266}{591}a^{6}-\frac{10}{197}a^{5}-\frac{163}{591}a^{4}+\frac{223}{591}a^{3}-\frac{227}{591}a^{2}+\frac{229}{591}a+\frac{1}{3}$, $\frac{1}{591}a^{39}+\frac{58}{591}a^{32}+\frac{28}{591}a^{31}-\frac{12}{197}a^{30}-\frac{11}{591}a^{29}+\frac{6}{197}a^{28}-\frac{17}{197}a^{27}-\frac{68}{197}a^{26}-\frac{37}{197}a^{25}+\frac{248}{591}a^{24}-\frac{170}{591}a^{23}-\frac{2}{591}a^{22}+\frac{89}{591}a^{21}-\frac{103}{591}a^{20}+\frac{81}{197}a^{19}+\frac{48}{197}a^{18}+\frac{203}{591}a^{17}-\frac{17}{591}a^{16}+\frac{98}{197}a^{15}+\frac{67}{591}a^{14}-\frac{4}{197}a^{13}-\frac{31}{591}a^{12}-\frac{272}{591}a^{11}+\frac{82}{197}a^{10}+\frac{1}{591}a^{9}+\frac{130}{591}a^{8}+\frac{260}{591}a^{7}-\frac{82}{197}a^{6}+\frac{25}{197}a^{5}-\frac{74}{197}a^{4}-\frac{77}{197}a^{3}+\frac{284}{591}a^{2}-\frac{277}{591}a-\frac{1}{3}$, $\frac{1}{49053}a^{40}+\frac{29}{49053}a^{39}+\frac{11}{16351}a^{38}-\frac{7}{49053}a^{37}+\frac{28}{49053}a^{36}+\frac{25}{49053}a^{35}+\frac{28}{49053}a^{34}+\frac{11}{16351}a^{33}-\frac{4219}{49053}a^{32}+\frac{2797}{49053}a^{31}-\frac{8120}{49053}a^{30}+\frac{19915}{49053}a^{29}-\frac{4060}{16351}a^{28}-\frac{7241}{16351}a^{27}-\frac{13772}{49053}a^{26}-\frac{9074}{49053}a^{25}+\frac{3917}{16351}a^{24}-\frac{9445}{49053}a^{23}+\frac{14098}{49053}a^{22}-\frac{10579}{49053}a^{21}-\frac{1876}{49053}a^{20}-\frac{18703}{49053}a^{19}+\frac{21076}{49053}a^{18}+\frac{1051}{16351}a^{17}-\frac{14131}{49053}a^{16}+\frac{22900}{49053}a^{15}-\frac{23756}{49053}a^{14}+\frac{19724}{49053}a^{13}-\frac{7519}{16351}a^{12}+\frac{18791}{49053}a^{11}-\frac{4}{249}a^{10}-\frac{20513}{49053}a^{9}-\frac{20275}{49053}a^{8}+\frac{7144}{16351}a^{7}+\frac{3169}{49053}a^{6}+\frac{11129}{49053}a^{5}-\frac{3539}{49053}a^{4}-\frac{14960}{49053}a^{3}+\frac{8512}{49053}a^{2}+\frac{9239}{49053}a+\frac{5}{83}$, $\frac{1}{49053}a^{41}+\frac{22}{49053}a^{39}+\frac{32}{49053}a^{38}-\frac{6}{16351}a^{37}-\frac{40}{49053}a^{36}-\frac{11}{16351}a^{35}-\frac{32}{49053}a^{34}-\frac{10}{16351}a^{33}-\frac{2357}{16351}a^{32}+\frac{3976}{49053}a^{31}-\frac{992}{49053}a^{30}-\frac{89}{591}a^{29}-\frac{14945}{49053}a^{28}+\frac{15358}{49053}a^{27}+\frac{3202}{49053}a^{26}-\frac{15935}{49053}a^{25}-\frac{6545}{16351}a^{24}-\frac{11461}{49053}a^{23}+\frac{5692}{16351}a^{22}+\frac{655}{16351}a^{21}-\frac{5336}{16351}a^{20}+\frac{6300}{16351}a^{19}+\frac{2852}{16351}a^{18}+\frac{1054}{16351}a^{17}+\frac{1473}{16351}a^{16}+\frac{7320}{16351}a^{15}+\frac{7879}{49053}a^{14}+\frac{21805}{49053}a^{13}+\frac{395}{49053}a^{12}+\frac{2932}{16351}a^{11}-\frac{991}{16351}a^{10}+\frac{2875}{16351}a^{9}+\frac{22514}{49053}a^{8}+\frac{14599}{49053}a^{7}+\frac{5695}{16351}a^{6}-\frac{10631}{49053}a^{5}+\frac{976}{16351}a^{4}-\frac{21452}{49053}a^{3}+\frac{4046}{16351}a^{2}-\frac{4606}{16351}a-\frac{103}{249}$, $\frac{1}{21534267}a^{42}-\frac{2}{7178089}a^{41}+\frac{176}{21534267}a^{40}-\frac{16135}{21534267}a^{39}+\frac{1054}{21534267}a^{38}+\frac{4634}{21534267}a^{37}-\frac{5355}{7178089}a^{36}-\frac{5744}{7178089}a^{35}-\frac{10715}{21534267}a^{34}-\frac{2429}{7178089}a^{33}+\frac{1756283}{21534267}a^{32}-\frac{579901}{21534267}a^{31}-\frac{1656706}{21534267}a^{30}-\frac{7404541}{21534267}a^{29}+\frac{4966999}{21534267}a^{28}+\frac{1416811}{7178089}a^{27}+\frac{1746560}{7178089}a^{26}-\frac{2820554}{7178089}a^{25}+\frac{148933}{21534267}a^{24}+\frac{2057054}{21534267}a^{23}+\frac{34621}{86483}a^{22}+\frac{2382314}{21534267}a^{21}-\frac{8396102}{21534267}a^{20}-\frac{2753003}{7178089}a^{19}+\frac{4442281}{21534267}a^{18}+\frac{5815877}{21534267}a^{17}-\frac{3424151}{21534267}a^{16}+\frac{2689666}{21534267}a^{15}-\frac{3742159}{21534267}a^{14}+\frac{1452108}{7178089}a^{13}+\frac{2815582}{21534267}a^{12}-\frac{139394}{21534267}a^{11}+\frac{8848942}{21534267}a^{10}+\frac{257618}{7178089}a^{9}-\frac{3102078}{7178089}a^{8}-\frac{388526}{7178089}a^{7}+\frac{738885}{7178089}a^{6}-\frac{885878}{21534267}a^{5}-\frac{247906}{7178089}a^{4}+\frac{7587850}{21534267}a^{3}+\frac{10222315}{21534267}a^{2}-\frac{92656}{21534267}a+\frac{20192}{109311}$, $\frac{1}{1022382394359}a^{43}+\frac{18515}{1022382394359}a^{42}+\frac{501455}{1022382394359}a^{41}+\frac{1243617}{340794131453}a^{40}+\frac{88410330}{340794131453}a^{39}-\frac{337660436}{1022382394359}a^{38}+\frac{136685948}{340794131453}a^{37}+\frac{455856898}{1022382394359}a^{36}-\frac{340090400}{1022382394359}a^{35}-\frac{5988171}{340794131453}a^{34}+\frac{100067381}{1022382394359}a^{33}+\frac{49098147717}{340794131453}a^{32}-\frac{120081598709}{1022382394359}a^{31}+\frac{2798534901}{340794131453}a^{30}+\frac{478980741439}{1022382394359}a^{29}-\frac{476480382272}{1022382394359}a^{28}+\frac{447560823590}{1022382394359}a^{27}-\frac{290717647823}{1022382394359}a^{26}-\frac{133383683896}{1022382394359}a^{25}-\frac{386788360343}{1022382394359}a^{24}-\frac{15531017920}{1022382394359}a^{23}+\frac{480043414285}{1022382394359}a^{22}-\frac{466277710057}{1022382394359}a^{21}+\frac{117526599584}{1022382394359}a^{20}-\frac{24594582855}{340794131453}a^{19}-\frac{91184523535}{340794131453}a^{18}-\frac{120008502092}{340794131453}a^{17}-\frac{11238903816}{340794131453}a^{16}-\frac{30269337359}{340794131453}a^{15}-\frac{77261448956}{1022382394359}a^{14}+\frac{450234916376}{1022382394359}a^{13}-\frac{1693038539}{4242250599}a^{12}+\frac{168519642659}{1022382394359}a^{11}-\frac{75935231130}{340794131453}a^{10}-\frac{426871526732}{1022382394359}a^{9}-\frac{146496197221}{340794131453}a^{8}+\frac{189347226652}{1022382394359}a^{7}-\frac{291417588770}{1022382394359}a^{6}-\frac{156149123920}{340794131453}a^{5}-\frac{98525967802}{340794131453}a^{4}-\frac{294363528529}{1022382394359}a^{3}+\frac{12808478123}{1022382394359}a^{2}+\frac{2170906724}{5189758347}a-\frac{3596653}{8781317}$, $\frac{1}{13\!\cdots\!21}a^{44}+\frac{35\!\cdots\!74}{13\!\cdots\!21}a^{43}-\frac{29\!\cdots\!95}{13\!\cdots\!21}a^{42}-\frac{31\!\cdots\!57}{13\!\cdots\!21}a^{41}+\frac{13\!\cdots\!87}{13\!\cdots\!21}a^{40}+\frac{29\!\cdots\!19}{13\!\cdots\!21}a^{39}+\frac{36\!\cdots\!05}{13\!\cdots\!21}a^{38}-\frac{73\!\cdots\!51}{45\!\cdots\!07}a^{37}-\frac{10\!\cdots\!03}{13\!\cdots\!21}a^{36}-\frac{11\!\cdots\!23}{45\!\cdots\!07}a^{35}+\frac{55\!\cdots\!90}{13\!\cdots\!21}a^{34}+\frac{31\!\cdots\!67}{45\!\cdots\!07}a^{33}-\frac{16\!\cdots\!74}{13\!\cdots\!21}a^{32}-\frac{61\!\cdots\!25}{13\!\cdots\!21}a^{31}+\frac{36\!\cdots\!38}{45\!\cdots\!07}a^{30}+\frac{21\!\cdots\!72}{45\!\cdots\!07}a^{29}-\frac{17\!\cdots\!36}{45\!\cdots\!07}a^{28}+\frac{69\!\cdots\!78}{13\!\cdots\!21}a^{27}+\frac{83\!\cdots\!86}{45\!\cdots\!07}a^{26}-\frac{65\!\cdots\!35}{13\!\cdots\!21}a^{25}+\frac{47\!\cdots\!96}{13\!\cdots\!21}a^{24}+\frac{28\!\cdots\!74}{13\!\cdots\!21}a^{23}+\frac{92\!\cdots\!62}{45\!\cdots\!07}a^{22}-\frac{23\!\cdots\!38}{13\!\cdots\!21}a^{21}-\frac{35\!\cdots\!48}{13\!\cdots\!21}a^{20}-\frac{15\!\cdots\!30}{45\!\cdots\!07}a^{19}-\frac{18\!\cdots\!08}{45\!\cdots\!07}a^{18}-\frac{66\!\cdots\!33}{13\!\cdots\!21}a^{17}+\frac{30\!\cdots\!00}{13\!\cdots\!21}a^{16}-\frac{40\!\cdots\!45}{45\!\cdots\!07}a^{15}+\frac{67\!\cdots\!13}{13\!\cdots\!21}a^{14}-\frac{48\!\cdots\!40}{13\!\cdots\!21}a^{13}-\frac{70\!\cdots\!88}{45\!\cdots\!07}a^{12}+\frac{15\!\cdots\!46}{13\!\cdots\!21}a^{11}-\frac{14\!\cdots\!05}{13\!\cdots\!21}a^{10}-\frac{30\!\cdots\!37}{54\!\cdots\!29}a^{9}+\frac{12\!\cdots\!53}{45\!\cdots\!07}a^{8}-\frac{33\!\cdots\!24}{13\!\cdots\!21}a^{7}+\frac{13\!\cdots\!83}{45\!\cdots\!07}a^{6}-\frac{56\!\cdots\!33}{13\!\cdots\!21}a^{5}-\frac{12\!\cdots\!16}{13\!\cdots\!21}a^{4}+\frac{37\!\cdots\!97}{13\!\cdots\!21}a^{3}+\frac{47\!\cdots\!32}{13\!\cdots\!21}a^{2}-\frac{10\!\cdots\!04}{69\!\cdots\!93}a+\frac{96\!\cdots\!15}{35\!\cdots\!69}$ Copy content Toggle raw display

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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321)
 
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 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321, 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 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321);
 
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 - 194*x^43 + 836*x^42 + 16500*x^41 - 76882*x^40 - 811248*x^39 + 4130184*x^38 + 25559474*x^37 - 145049280*x^36 - 538597921*x^35 + 3530594867*x^34 + 7615564821*x^33 - 61610572814*x^32 - 68317895227*x^31 + 786429242050*x^30 + 287086043611*x^29 - 7425134094508*x^28 + 1391788361320*x^27 + 52087220030130*x^26 - 32183862662002*x^25 - 271116028008606*x^24 + 264064430436474*x^23 + 1039304723755570*x^22 - 1345885450620938*x^21 - 2889398096911074*x^20 + 4691240971935913*x^19 + 5661912712400497*x^18 - 11481658937434788*x^17 - 7388527635358822*x^16 + 19748315799639590*x^15 + 5543796759733632*x^14 - 23515522924776395*x^13 - 895060410571712*x^12 + 18817979236586386*x^11 - 2391413888018270*x^10 - 9652129249107031*x^9 + 2328285637187904*x^8 + 2951322314183680*x^7 - 960094366581828*x^6 - 477351520888160*x^5 + 190413802657905*x^4 + 31093339218140*x^3 - 15654659537914*x^2 - 188842256201*x + 306884845321);
 
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 $C_{45}$ is not computed

Intermediate fields

3.3.5329.1, \(\Q(\zeta_{11})^+\), 9.9.806460091894081.1, 15.15.13487790856470777216989713088929.1

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 $45$ $15^{3}$ $45$ $15^{3}$ R $45$ $15^{3}$ $45$ ${\href{/padicField/23.9.0.1}{9} }^{5}$ $45$ $45$ $45$ $45$ ${\href{/padicField/43.3.0.1}{3} }^{15}$ $45$ $45$ $45$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display Deg $45$$5$$9$$36$
\(73\) Copy content Toggle raw display Deg $45$$9$$5$$40$