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Properties

Label 45.45.999...561.1
Degree $45$
Signature $[45, 0]$
Discriminant $9.995\times 10^{100}$
Root discriminant \(175.57\)
Ramified primes $19,31$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919)
 
gp: K = bnfinit(y^45 - 12*y^44 - 68*y^43 + 1390*y^42 - 278*y^41 - 68270*y^40 + 176199*y^39 + 1808389*y^38 - 8154574*y^37 - 26005805*y^36 + 196321268*y^35 + 127187382*y^34 - 2913805970*y^33 + 2272631340*y^32 + 27669670927*y^31 - 52510157887*y^30 - 162002621792*y^29 + 526806359978*y^28 + 468745083598*y^27 - 3175101889498*y^26 + 532391340063*y^25 + 12135569776697*y^24 - 10651958891479*y^23 - 28871893561060*y^22 + 45196495080913*y^21 + 38272655903792*y^20 - 105664143365844*y^19 - 13349385343870*y^18 + 153132315545987*y^17 - 41028745873427*y^16 - 141811047613612*y^15 + 76543731297893*y^14 + 83515595010770*y^13 - 65499266094722*y^12 - 29808531214812*y^11 + 32918818373161*y^10 + 5404646072895*y^9 - 10084991417837*y^8 - 16286468525*y^7 + 1821205338619*y^6 - 180698614131*y^5 - 173451418339*y^4 + 28592722199*y^3 + 6396948952*y^2 - 1303357845*y - 3078919, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919)
 

\( x^{45} - 12 x^{44} - 68 x^{43} + 1390 x^{42} - 278 x^{41} - 68270 x^{40} + 176199 x^{39} + \cdots - 3078919 \) 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:   \(999\!\cdots\!561\) \(\medspace = 19^{30}\cdot 31^{42}\) 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:  \(175.57\)
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:  $19^{2/3}31^{14/15}\approx 175.5658249685855$
Ramified primes:   \(19\), \(31\) 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:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(514,·)$, $\chi_{589}(134,·)$, $\chi_{589}(391,·)$, $\chi_{589}(267,·)$, $\chi_{589}(524,·)$, $\chi_{589}(258,·)$, $\chi_{589}(400,·)$, $\chi_{589}(273,·)$, $\chi_{589}(20,·)$, $\chi_{589}(410,·)$, $\chi_{589}(286,·)$, $\chi_{589}(543,·)$, $\chi_{589}(163,·)$, $\chi_{589}(39,·)$, $\chi_{589}(552,·)$, $\chi_{589}(7,·)$, $\chi_{589}(45,·)$, $\chi_{589}(49,·)$, $\chi_{589}(562,·)$, $\chi_{589}(438,·)$, $\chi_{589}(311,·)$, $\chi_{589}(87,·)$, $\chi_{589}(159,·)$, $\chi_{589}(444,·)$, $\chi_{589}(191,·)$, $\chi_{589}(64,·)$, $\chi_{589}(577,·)$, $\chi_{589}(324,·)$, $\chi_{589}(140,·)$, $\chi_{589}(330,·)$, $\chi_{589}(419,·)$, $\chi_{589}(343,·)$, $\chi_{589}(349,·)$, $\chi_{589}(144,·)$, $\chi_{589}(315,·)$, $\chi_{589}(448,·)$, $\chi_{589}(102,·)$, $\chi_{589}(235,·)$, $\chi_{589}(125,·)$, $\chi_{589}(467,·)$, $\chi_{589}(501,·)$, $\chi_{589}(505,·)$, $\chi_{589}(121,·)$, $\chi_{589}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{5}a^{39}+\frac{1}{5}a^{38}+\frac{1}{5}a^{37}+\frac{1}{5}a^{36}+\frac{2}{5}a^{35}+\frac{1}{5}a^{33}-\frac{1}{5}a^{32}+\frac{1}{5}a^{31}+\frac{2}{5}a^{28}+\frac{2}{5}a^{26}-\frac{1}{5}a^{25}-\frac{2}{5}a^{22}+\frac{2}{5}a^{21}-\frac{2}{5}a^{20}-\frac{2}{5}a^{19}-\frac{1}{5}a^{17}-\frac{1}{5}a^{15}-\frac{1}{5}a^{14}-\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{40}+\frac{1}{5}a^{36}-\frac{2}{5}a^{35}+\frac{1}{5}a^{34}-\frac{2}{5}a^{33}+\frac{2}{5}a^{32}-\frac{1}{5}a^{31}+\frac{2}{5}a^{29}-\frac{2}{5}a^{28}+\frac{2}{5}a^{27}+\frac{2}{5}a^{26}+\frac{1}{5}a^{25}-\frac{2}{5}a^{23}-\frac{1}{5}a^{22}+\frac{1}{5}a^{21}+\frac{2}{5}a^{19}-\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{41}+\frac{1}{5}a^{37}-\frac{2}{5}a^{36}+\frac{1}{5}a^{35}-\frac{2}{5}a^{34}+\frac{2}{5}a^{33}-\frac{1}{5}a^{32}+\frac{2}{5}a^{30}-\frac{2}{5}a^{29}+\frac{2}{5}a^{28}+\frac{2}{5}a^{27}+\frac{1}{5}a^{26}-\frac{2}{5}a^{24}-\frac{1}{5}a^{23}+\frac{1}{5}a^{22}+\frac{2}{5}a^{20}-\frac{1}{5}a^{19}+\frac{1}{5}a^{18}-\frac{1}{5}a^{17}-\frac{2}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{42}+\frac{1}{5}a^{38}-\frac{2}{5}a^{37}+\frac{1}{5}a^{36}-\frac{2}{5}a^{35}+\frac{2}{5}a^{34}-\frac{1}{5}a^{33}+\frac{2}{5}a^{31}-\frac{2}{5}a^{30}+\frac{2}{5}a^{29}+\frac{2}{5}a^{28}+\frac{1}{5}a^{27}-\frac{2}{5}a^{25}-\frac{1}{5}a^{24}+\frac{1}{5}a^{23}+\frac{2}{5}a^{21}-\frac{1}{5}a^{20}+\frac{1}{5}a^{19}-\frac{1}{5}a^{18}-\frac{2}{5}a^{15}+\frac{2}{5}a^{14}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{962545}a^{43}+\frac{78469}{962545}a^{42}+\frac{13438}{962545}a^{41}+\frac{7167}{962545}a^{40}-\frac{9391}{192509}a^{39}-\frac{471284}{962545}a^{38}-\frac{68393}{192509}a^{37}-\frac{441683}{962545}a^{36}-\frac{39624}{962545}a^{35}-\frac{327672}{962545}a^{34}+\frac{118222}{962545}a^{33}-\frac{97676}{962545}a^{32}+\frac{479673}{962545}a^{31}-\frac{79324}{192509}a^{30}+\frac{35613}{962545}a^{29}+\frac{195429}{962545}a^{28}+\frac{73139}{962545}a^{27}+\frac{62633}{962545}a^{26}-\frac{22906}{962545}a^{25}-\frac{48199}{962545}a^{24}-\frac{430023}{962545}a^{23}-\frac{16978}{192509}a^{22}-\frac{58918}{962545}a^{21}-\frac{42822}{192509}a^{20}-\frac{202204}{962545}a^{19}-\frac{168908}{962545}a^{18}-\frac{6755}{192509}a^{17}-\frac{238359}{962545}a^{16}+\frac{34096}{192509}a^{15}+\frac{382868}{962545}a^{14}+\frac{403944}{962545}a^{13}+\frac{13754}{192509}a^{12}-\frac{61531}{962545}a^{11}+\frac{361207}{962545}a^{10}+\frac{342012}{962545}a^{9}-\frac{35237}{962545}a^{8}+\frac{205486}{962545}a^{7}+\frac{84271}{962545}a^{6}+\frac{249694}{962545}a^{5}+\frac{15878}{962545}a^{4}+\frac{202902}{962545}a^{3}+\frac{178984}{962545}a^{2}-\frac{407646}{962545}a-\frac{275706}{962545}$, $\frac{1}{25\!\cdots\!15}a^{44}-\frac{11\!\cdots\!36}{25\!\cdots\!15}a^{43}+\frac{19\!\cdots\!37}{25\!\cdots\!15}a^{42}-\frac{10\!\cdots\!71}{50\!\cdots\!03}a^{41}-\frac{94\!\cdots\!35}{50\!\cdots\!03}a^{40}+\frac{12\!\cdots\!11}{25\!\cdots\!15}a^{39}+\frac{37\!\cdots\!24}{80\!\cdots\!65}a^{38}+\frac{14\!\cdots\!37}{25\!\cdots\!15}a^{37}+\frac{69\!\cdots\!34}{25\!\cdots\!15}a^{36}+\frac{89\!\cdots\!98}{25\!\cdots\!15}a^{35}+\frac{17\!\cdots\!84}{25\!\cdots\!15}a^{34}-\frac{10\!\cdots\!89}{25\!\cdots\!15}a^{33}-\frac{10\!\cdots\!56}{50\!\cdots\!03}a^{32}+\frac{28\!\cdots\!43}{25\!\cdots\!15}a^{31}+\frac{97\!\cdots\!01}{25\!\cdots\!15}a^{30}-\frac{99\!\cdots\!14}{25\!\cdots\!15}a^{29}-\frac{70\!\cdots\!12}{25\!\cdots\!15}a^{28}-\frac{36\!\cdots\!32}{25\!\cdots\!15}a^{27}-\frac{22\!\cdots\!73}{25\!\cdots\!15}a^{26}-\frac{67\!\cdots\!97}{25\!\cdots\!15}a^{25}-\frac{12\!\cdots\!03}{25\!\cdots\!15}a^{24}+\frac{76\!\cdots\!01}{25\!\cdots\!15}a^{23}-\frac{99\!\cdots\!51}{50\!\cdots\!03}a^{22}+\frac{11\!\cdots\!03}{25\!\cdots\!15}a^{21}+\frac{67\!\cdots\!23}{25\!\cdots\!15}a^{20}+\frac{10\!\cdots\!38}{25\!\cdots\!15}a^{19}+\frac{68\!\cdots\!89}{25\!\cdots\!15}a^{18}+\frac{20\!\cdots\!23}{25\!\cdots\!15}a^{17}-\frac{15\!\cdots\!89}{50\!\cdots\!03}a^{16}+\frac{21\!\cdots\!63}{50\!\cdots\!03}a^{15}+\frac{54\!\cdots\!31}{25\!\cdots\!15}a^{14}-\frac{11\!\cdots\!39}{25\!\cdots\!15}a^{13}+\frac{91\!\cdots\!18}{25\!\cdots\!15}a^{12}+\frac{51\!\cdots\!41}{25\!\cdots\!15}a^{11}+\frac{18\!\cdots\!12}{50\!\cdots\!03}a^{10}+\frac{53\!\cdots\!79}{25\!\cdots\!15}a^{9}-\frac{28\!\cdots\!14}{25\!\cdots\!15}a^{8}-\frac{11\!\cdots\!69}{25\!\cdots\!15}a^{7}+\frac{22\!\cdots\!89}{25\!\cdots\!15}a^{6}+\frac{39\!\cdots\!04}{50\!\cdots\!03}a^{5}-\frac{23\!\cdots\!87}{25\!\cdots\!15}a^{4}-\frac{10\!\cdots\!02}{25\!\cdots\!15}a^{3}-\frac{11\!\cdots\!12}{25\!\cdots\!15}a^{2}+\frac{31\!\cdots\!43}{25\!\cdots\!15}a+\frac{12\!\cdots\!81}{50\!\cdots\!03}$ 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 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919)
 
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 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919, 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 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919);
 
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 - 12*x^44 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919);
 
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

$C_3\times C_{15}$ (as 45T2):

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)
 
An abelian group of order 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

3.3.346921.2, 3.3.361.1, 3.3.961.1, 3.3.346921.1, 5.5.923521.1, 9.9.41753392563387961.1, 15.15.4640873420279330256301704361074121.2, 15.15.4829212716211581952447142935561.1, \(\Q(\zeta_{31})^+\), 15.15.4640873420279330256301704361074121.1

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 $15^{3}$ $15^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ R $15^{3}$ $15^{3}$ R ${\href{/padicField/37.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$

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
\(19\) Copy content Toggle raw display Deg $45$$3$$15$$30$
\(31\) Copy content Toggle raw display 31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} + 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$