Properties

Label 28.0.502...125.1
Degree $28$
Signature $[0, 14]$
Discriminant $5.020\times 10^{52}$
Root discriminant \(76.24\)
Ramified primes $5,29$
Class number $8992$ (GRH)
Class group [2, 2, 2, 2, 562] (GRH)
Galois group $C_{28}$ (as 28T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111)
 
gp: K = bnfinit(y^28 - y^27 + 18*y^26 - 23*y^25 + 29*y^24 - 149*y^23 - 795*y^22 - 8*y^21 - 2927*y^20 + 667*y^19 + 13846*y^18 - 647*y^17 + 109320*y^16 + 98963*y^15 + 313397*y^14 + 494999*y^13 + 789626*y^12 + 291678*y^11 + 1921156*y^10 - 425111*y^9 + 1044768*y^8 + 1156338*y^7 - 1030264*y^6 + 642721*y^5 + 1780368*y^4 - 4481806*y^3 + 897127*y^2 + 1339909*y + 1148111, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111)
 

\( x^{28} - x^{27} + 18 x^{26} - 23 x^{25} + 29 x^{24} - 149 x^{23} - 795 x^{22} - 8 x^{21} - 2927 x^{20} + 667 x^{19} + 13846 x^{18} - 647 x^{17} + 109320 x^{16} + 98963 x^{15} + \cdots + 1148111 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(50201655190081835380839261671426578388690948486328125\) \(\medspace = 5^{21}\cdot 29^{26}\) 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:  \(76.24\)
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:  $5^{3/4}29^{13/14}\approx 76.23747453486138$
Ramified primes:   \(5\), \(29\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(67,·)$, $\chi_{145}(136,·)$, $\chi_{145}(138,·)$, $\chi_{145}(139,·)$, $\chi_{145}(13,·)$, $\chi_{145}(141,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(22,·)$, $\chi_{145}(24,·)$, $\chi_{145}(28,·)$, $\chi_{145}(93,·)$, $\chi_{145}(94,·)$, $\chi_{145}(33,·)$, $\chi_{145}(36,·)$, $\chi_{145}(38,·)$, $\chi_{145}(92,·)$, $\chi_{145}(42,·)$, $\chi_{145}(111,·)$, $\chi_{145}(49,·)$, $\chi_{145}(54,·)$, $\chi_{145}(57,·)$, $\chi_{145}(122,·)$, $\chi_{145}(59,·)$, $\chi_{145}(74,·)$, $\chi_{145}(62,·)$, $\chi_{145}(63,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{1035241}a^{26}+\frac{167675}{1035241}a^{25}-\frac{8607}{1035241}a^{24}-\frac{49857}{1035241}a^{23}-\frac{327836}{1035241}a^{22}-\frac{226077}{1035241}a^{21}+\frac{22454}{1035241}a^{20}+\frac{89303}{1035241}a^{19}-\frac{285793}{1035241}a^{18}+\frac{14038}{1035241}a^{17}-\frac{69319}{1035241}a^{16}-\frac{323357}{1035241}a^{15}+\frac{64696}{1035241}a^{14}+\frac{455141}{1035241}a^{13}-\frac{285930}{1035241}a^{12}-\frac{332747}{1035241}a^{11}+\frac{504087}{1035241}a^{10}+\frac{50072}{1035241}a^{9}-\frac{71359}{1035241}a^{8}-\frac{479530}{1035241}a^{7}+\frac{431797}{1035241}a^{6}+\frac{22510}{1035241}a^{5}+\frac{37288}{1035241}a^{4}+\frac{214708}{1035241}a^{3}-\frac{58931}{1035241}a^{2}-\frac{343309}{1035241}a-\frac{485626}{1035241}$, $\frac{1}{26\!\cdots\!41}a^{27}-\frac{71\!\cdots\!17}{26\!\cdots\!41}a^{26}-\frac{11\!\cdots\!44}{26\!\cdots\!41}a^{25}-\frac{54\!\cdots\!88}{26\!\cdots\!41}a^{24}-\frac{87\!\cdots\!83}{26\!\cdots\!41}a^{23}-\frac{84\!\cdots\!48}{26\!\cdots\!41}a^{22}-\frac{57\!\cdots\!27}{26\!\cdots\!41}a^{21}+\frac{10\!\cdots\!26}{26\!\cdots\!41}a^{20}+\frac{11\!\cdots\!26}{26\!\cdots\!41}a^{19}+\frac{66\!\cdots\!56}{26\!\cdots\!41}a^{18}+\frac{11\!\cdots\!99}{26\!\cdots\!41}a^{17}+\frac{11\!\cdots\!63}{26\!\cdots\!41}a^{16}-\frac{52\!\cdots\!41}{26\!\cdots\!41}a^{15}+\frac{11\!\cdots\!21}{26\!\cdots\!41}a^{14}-\frac{61\!\cdots\!98}{26\!\cdots\!41}a^{13}+\frac{91\!\cdots\!19}{26\!\cdots\!41}a^{12}-\frac{84\!\cdots\!48}{26\!\cdots\!41}a^{11}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{10}-\frac{99\!\cdots\!42}{26\!\cdots\!41}a^{9}+\frac{59\!\cdots\!48}{26\!\cdots\!41}a^{8}+\frac{86\!\cdots\!70}{26\!\cdots\!41}a^{7}+\frac{12\!\cdots\!48}{26\!\cdots\!41}a^{6}+\frac{10\!\cdots\!12}{26\!\cdots\!41}a^{5}-\frac{12\!\cdots\!98}{26\!\cdots\!41}a^{4}+\frac{44\!\cdots\!03}{26\!\cdots\!41}a^{3}+\frac{81\!\cdots\!77}{26\!\cdots\!41}a^{2}-\frac{12\!\cdots\!37}{26\!\cdots\!41}a+\frac{10\!\cdots\!31}{26\!\cdots\!41}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{562}$, which has order $8992$ (assuming GRH)

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:  $13$
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:   $\frac{42\!\cdots\!42}{15\!\cdots\!61}a^{27}-\frac{24\!\cdots\!90}{15\!\cdots\!61}a^{26}+\frac{75\!\cdots\!76}{15\!\cdots\!61}a^{25}-\frac{63\!\cdots\!12}{15\!\cdots\!61}a^{24}+\frac{10\!\cdots\!99}{15\!\cdots\!61}a^{23}-\frac{56\!\cdots\!85}{15\!\cdots\!61}a^{22}-\frac{36\!\cdots\!26}{15\!\cdots\!61}a^{21}-\frac{16\!\cdots\!24}{15\!\cdots\!61}a^{20}-\frac{13\!\cdots\!19}{15\!\cdots\!61}a^{19}-\frac{49\!\cdots\!68}{15\!\cdots\!61}a^{18}+\frac{55\!\cdots\!12}{15\!\cdots\!61}a^{17}+\frac{19\!\cdots\!95}{15\!\cdots\!61}a^{16}+\frac{47\!\cdots\!68}{15\!\cdots\!61}a^{15}+\frac{66\!\cdots\!97}{15\!\cdots\!61}a^{14}+\frac{16\!\cdots\!14}{15\!\cdots\!61}a^{13}+\frac{30\!\cdots\!33}{15\!\cdots\!61}a^{12}+\frac{50\!\cdots\!09}{15\!\cdots\!61}a^{11}+\frac{38\!\cdots\!27}{15\!\cdots\!61}a^{10}+\frac{10\!\cdots\!88}{15\!\cdots\!61}a^{9}+\frac{37\!\cdots\!04}{15\!\cdots\!61}a^{8}+\frac{51\!\cdots\!52}{15\!\cdots\!61}a^{7}+\frac{85\!\cdots\!64}{15\!\cdots\!61}a^{6}-\frac{19\!\cdots\!20}{15\!\cdots\!61}a^{5}-\frac{36\!\cdots\!67}{15\!\cdots\!61}a^{4}+\frac{56\!\cdots\!43}{15\!\cdots\!61}a^{3}-\frac{16\!\cdots\!67}{15\!\cdots\!61}a^{2}-\frac{12\!\cdots\!44}{15\!\cdots\!61}a+\frac{40\!\cdots\!07}{15\!\cdots\!61}$, $\frac{62\!\cdots\!60}{26\!\cdots\!41}a^{27}-\frac{44\!\cdots\!95}{26\!\cdots\!41}a^{26}+\frac{19\!\cdots\!05}{26\!\cdots\!41}a^{25}-\frac{96\!\cdots\!85}{26\!\cdots\!41}a^{24}+\frac{36\!\cdots\!55}{26\!\cdots\!41}a^{23}-\frac{50\!\cdots\!50}{26\!\cdots\!41}a^{22}+\frac{11\!\cdots\!95}{26\!\cdots\!41}a^{21}+\frac{11\!\cdots\!80}{26\!\cdots\!41}a^{20}-\frac{10\!\cdots\!70}{26\!\cdots\!41}a^{19}+\frac{13\!\cdots\!10}{26\!\cdots\!41}a^{18}-\frac{46\!\cdots\!20}{26\!\cdots\!41}a^{17}-\frac{43\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{19\!\cdots\!60}{26\!\cdots\!41}a^{15}-\frac{48\!\cdots\!45}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!70}{26\!\cdots\!41}a^{13}-\frac{18\!\cdots\!10}{26\!\cdots\!41}a^{12}+\frac{26\!\cdots\!25}{26\!\cdots\!41}a^{11}+\frac{30\!\cdots\!70}{26\!\cdots\!41}a^{10}+\frac{97\!\cdots\!35}{26\!\cdots\!41}a^{9}-\frac{44\!\cdots\!45}{26\!\cdots\!41}a^{8}+\frac{28\!\cdots\!40}{26\!\cdots\!41}a^{7}-\frac{18\!\cdots\!50}{26\!\cdots\!41}a^{6}+\frac{99\!\cdots\!31}{26\!\cdots\!41}a^{5}+\frac{49\!\cdots\!30}{26\!\cdots\!41}a^{4}-\frac{32\!\cdots\!00}{26\!\cdots\!41}a^{3}+\frac{53\!\cdots\!75}{26\!\cdots\!41}a^{2}+\frac{10\!\cdots\!30}{26\!\cdots\!41}a-\frac{81\!\cdots\!00}{26\!\cdots\!41}$, $\frac{65\!\cdots\!32}{26\!\cdots\!41}a^{27}-\frac{10\!\cdots\!40}{26\!\cdots\!41}a^{26}+\frac{11\!\cdots\!15}{26\!\cdots\!41}a^{25}-\frac{22\!\cdots\!97}{26\!\cdots\!41}a^{24}+\frac{21\!\cdots\!30}{26\!\cdots\!41}a^{23}-\frac{11\!\cdots\!95}{26\!\cdots\!41}a^{22}-\frac{44\!\cdots\!89}{26\!\cdots\!41}a^{21}+\frac{33\!\cdots\!20}{26\!\cdots\!41}a^{20}-\frac{14\!\cdots\!45}{26\!\cdots\!41}a^{19}+\frac{23\!\cdots\!86}{26\!\cdots\!41}a^{18}+\frac{97\!\cdots\!02}{26\!\cdots\!41}a^{17}-\frac{32\!\cdots\!85}{26\!\cdots\!41}a^{16}+\frac{63\!\cdots\!15}{26\!\cdots\!41}a^{15}+\frac{73\!\cdots\!86}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!51}{26\!\cdots\!41}a^{13}+\frac{45\!\cdots\!32}{26\!\cdots\!41}a^{12}+\frac{12\!\cdots\!89}{26\!\cdots\!41}a^{11}-\frac{60\!\cdots\!15}{26\!\cdots\!41}a^{10}+\frac{44\!\cdots\!78}{26\!\cdots\!41}a^{9}-\frac{16\!\cdots\!47}{26\!\cdots\!41}a^{8}+\frac{15\!\cdots\!26}{26\!\cdots\!41}a^{7}-\frac{61\!\cdots\!63}{26\!\cdots\!41}a^{6}-\frac{10\!\cdots\!57}{26\!\cdots\!41}a^{5}+\frac{60\!\cdots\!80}{26\!\cdots\!41}a^{4}+\frac{17\!\cdots\!43}{26\!\cdots\!41}a^{3}-\frac{55\!\cdots\!09}{26\!\cdots\!41}a^{2}-\frac{69\!\cdots\!87}{26\!\cdots\!41}a+\frac{24\!\cdots\!77}{26\!\cdots\!41}$, $\frac{16\!\cdots\!97}{26\!\cdots\!41}a^{27}-\frac{60\!\cdots\!70}{26\!\cdots\!41}a^{26}+\frac{38\!\cdots\!33}{26\!\cdots\!41}a^{25}-\frac{11\!\cdots\!37}{26\!\cdots\!41}a^{24}+\frac{22\!\cdots\!10}{26\!\cdots\!41}a^{23}-\frac{44\!\cdots\!50}{26\!\cdots\!41}a^{22}-\frac{48\!\cdots\!04}{26\!\cdots\!41}a^{21}+\frac{27\!\cdots\!52}{26\!\cdots\!41}a^{20}-\frac{79\!\cdots\!85}{26\!\cdots\!41}a^{19}+\frac{11\!\cdots\!36}{26\!\cdots\!41}a^{18}+\frac{40\!\cdots\!02}{26\!\cdots\!41}a^{17}-\frac{58\!\cdots\!65}{26\!\cdots\!41}a^{16}+\frac{22\!\cdots\!00}{26\!\cdots\!41}a^{15}-\frac{27\!\cdots\!49}{26\!\cdots\!41}a^{14}+\frac{62\!\cdots\!16}{26\!\cdots\!41}a^{13}-\frac{66\!\cdots\!83}{26\!\cdots\!41}a^{12}+\frac{12\!\cdots\!49}{26\!\cdots\!41}a^{11}-\frac{36\!\cdots\!43}{26\!\cdots\!41}a^{10}+\frac{67\!\cdots\!83}{26\!\cdots\!41}a^{9}-\frac{51\!\cdots\!92}{26\!\cdots\!41}a^{8}+\frac{12\!\cdots\!66}{26\!\cdots\!41}a^{7}-\frac{50\!\cdots\!98}{26\!\cdots\!41}a^{6}+\frac{26\!\cdots\!10}{26\!\cdots\!41}a^{5}-\frac{23\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{84\!\cdots\!87}{26\!\cdots\!41}a^{3}+\frac{26\!\cdots\!31}{26\!\cdots\!41}a^{2}+\frac{36\!\cdots\!98}{26\!\cdots\!41}a+\frac{87\!\cdots\!24}{26\!\cdots\!41}$, $\frac{62\!\cdots\!78}{26\!\cdots\!41}a^{27}+\frac{89\!\cdots\!75}{26\!\cdots\!41}a^{26}+\frac{15\!\cdots\!73}{26\!\cdots\!41}a^{25}-\frac{10\!\cdots\!28}{26\!\cdots\!41}a^{24}+\frac{87\!\cdots\!25}{26\!\cdots\!41}a^{23}-\frac{32\!\cdots\!60}{26\!\cdots\!41}a^{22}-\frac{54\!\cdots\!46}{26\!\cdots\!41}a^{21}-\frac{21\!\cdots\!59}{26\!\cdots\!41}a^{20}-\frac{55\!\cdots\!65}{26\!\cdots\!41}a^{19}+\frac{41\!\cdots\!94}{26\!\cdots\!41}a^{18}+\frac{15\!\cdots\!33}{26\!\cdots\!41}a^{17}+\frac{49\!\cdots\!75}{26\!\cdots\!41}a^{16}+\frac{16\!\cdots\!33}{26\!\cdots\!41}a^{15}+\frac{90\!\cdots\!14}{26\!\cdots\!41}a^{14}+\frac{80\!\cdots\!04}{26\!\cdots\!41}a^{13}+\frac{43\!\cdots\!53}{26\!\cdots\!41}a^{12}+\frac{11\!\cdots\!01}{26\!\cdots\!41}a^{11}+\frac{82\!\cdots\!65}{26\!\cdots\!41}a^{10}-\frac{23\!\cdots\!48}{26\!\cdots\!41}a^{9}-\frac{58\!\cdots\!13}{26\!\cdots\!41}a^{8}+\frac{51\!\cdots\!24}{26\!\cdots\!41}a^{7}-\frac{11\!\cdots\!02}{26\!\cdots\!41}a^{6}+\frac{61\!\cdots\!89}{26\!\cdots\!41}a^{5}+\frac{13\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{77\!\cdots\!58}{26\!\cdots\!41}a^{3}-\frac{30\!\cdots\!31}{26\!\cdots\!41}a^{2}-\frac{56\!\cdots\!78}{26\!\cdots\!41}a-\frac{37\!\cdots\!09}{26\!\cdots\!41}$, $\frac{84\!\cdots\!24}{26\!\cdots\!41}a^{27}-\frac{23\!\cdots\!70}{26\!\cdots\!41}a^{26}+\frac{16\!\cdots\!24}{26\!\cdots\!41}a^{25}-\frac{49\!\cdots\!84}{26\!\cdots\!41}a^{24}+\frac{70\!\cdots\!40}{26\!\cdots\!41}a^{23}-\frac{22\!\cdots\!25}{26\!\cdots\!41}a^{22}-\frac{16\!\cdots\!58}{26\!\cdots\!41}a^{21}+\frac{10\!\cdots\!59}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!65}{26\!\cdots\!41}a^{19}+\frac{60\!\cdots\!72}{26\!\cdots\!41}a^{18}-\frac{62\!\cdots\!31}{26\!\cdots\!41}a^{17}-\frac{20\!\cdots\!60}{26\!\cdots\!41}a^{16}+\frac{65\!\cdots\!27}{26\!\cdots\!41}a^{15}-\frac{13\!\cdots\!33}{26\!\cdots\!41}a^{14}+\frac{31\!\cdots\!02}{26\!\cdots\!41}a^{13}-\frac{22\!\cdots\!06}{26\!\cdots\!41}a^{12}+\frac{68\!\cdots\!08}{26\!\cdots\!41}a^{11}-\frac{13\!\cdots\!73}{26\!\cdots\!41}a^{10}+\frac{32\!\cdots\!06}{26\!\cdots\!41}a^{9}-\frac{12\!\cdots\!84}{26\!\cdots\!41}a^{8}+\frac{81\!\cdots\!62}{26\!\cdots\!41}a^{7}-\frac{38\!\cdots\!06}{26\!\cdots\!41}a^{6}+\frac{29\!\cdots\!53}{26\!\cdots\!41}a^{5}-\frac{16\!\cdots\!20}{26\!\cdots\!41}a^{4}-\frac{56\!\cdots\!64}{26\!\cdots\!41}a^{3}+\frac{21\!\cdots\!02}{26\!\cdots\!41}a^{2}+\frac{26\!\cdots\!31}{26\!\cdots\!41}a-\frac{14\!\cdots\!34}{26\!\cdots\!41}$, $\frac{84\!\cdots\!73}{26\!\cdots\!41}a^{27}-\frac{37\!\cdots\!00}{26\!\cdots\!41}a^{26}+\frac{22\!\cdots\!09}{26\!\cdots\!41}a^{25}-\frac{70\!\cdots\!53}{26\!\cdots\!41}a^{24}+\frac{15\!\cdots\!70}{26\!\cdots\!41}a^{23}-\frac{22\!\cdots\!25}{26\!\cdots\!41}a^{22}-\frac{32\!\cdots\!46}{26\!\cdots\!41}a^{21}+\frac{16\!\cdots\!93}{26\!\cdots\!41}a^{20}-\frac{60\!\cdots\!20}{26\!\cdots\!41}a^{19}+\frac{56\!\cdots\!64}{26\!\cdots\!41}a^{18}+\frac{46\!\cdots\!33}{26\!\cdots\!41}a^{17}-\frac{38\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{15\!\cdots\!73}{26\!\cdots\!41}a^{15}-\frac{14\!\cdots\!16}{26\!\cdots\!41}a^{14}+\frac{31\!\cdots\!14}{26\!\cdots\!41}a^{13}+\frac{15\!\cdots\!23}{26\!\cdots\!41}a^{12}+\frac{55\!\cdots\!41}{26\!\cdots\!41}a^{11}-\frac{22\!\cdots\!70}{26\!\cdots\!41}a^{10}+\frac{35\!\cdots\!77}{26\!\cdots\!41}a^{9}-\frac{50\!\cdots\!08}{26\!\cdots\!41}a^{8}+\frac{38\!\cdots\!04}{26\!\cdots\!41}a^{7}-\frac{12\!\cdots\!92}{26\!\cdots\!41}a^{6}+\frac{23\!\cdots\!57}{26\!\cdots\!41}a^{5}-\frac{62\!\cdots\!00}{26\!\cdots\!41}a^{4}-\frac{28\!\cdots\!23}{26\!\cdots\!41}a^{3}+\frac{44\!\cdots\!29}{26\!\cdots\!41}a^{2}+\frac{10\!\cdots\!67}{26\!\cdots\!41}a-\frac{39\!\cdots\!83}{26\!\cdots\!41}$, $\frac{25\!\cdots\!29}{26\!\cdots\!41}a^{27}-\frac{86\!\cdots\!51}{26\!\cdots\!41}a^{26}+\frac{45\!\cdots\!13}{26\!\cdots\!41}a^{25}-\frac{29\!\cdots\!11}{26\!\cdots\!41}a^{24}+\frac{58\!\cdots\!41}{26\!\cdots\!41}a^{23}-\frac{35\!\cdots\!12}{26\!\cdots\!41}a^{22}-\frac{22\!\cdots\!63}{26\!\cdots\!41}a^{21}-\frac{15\!\cdots\!74}{26\!\cdots\!41}a^{20}-\frac{84\!\cdots\!77}{26\!\cdots\!41}a^{19}-\frac{34\!\cdots\!06}{26\!\cdots\!41}a^{18}+\frac{31\!\cdots\!27}{26\!\cdots\!41}a^{17}+\frac{21\!\cdots\!05}{26\!\cdots\!41}a^{16}+\frac{29\!\cdots\!74}{26\!\cdots\!41}a^{15}+\frac{43\!\cdots\!17}{26\!\cdots\!41}a^{14}+\frac{11\!\cdots\!15}{26\!\cdots\!41}a^{13}+\frac{19\!\cdots\!91}{26\!\cdots\!41}a^{12}+\frac{32\!\cdots\!00}{26\!\cdots\!41}a^{11}+\frac{27\!\cdots\!38}{26\!\cdots\!41}a^{10}+\frac{65\!\cdots\!44}{26\!\cdots\!41}a^{9}+\frac{26\!\cdots\!63}{26\!\cdots\!41}a^{8}+\frac{43\!\cdots\!10}{26\!\cdots\!41}a^{7}+\frac{36\!\cdots\!04}{26\!\cdots\!41}a^{6}-\frac{57\!\cdots\!83}{26\!\cdots\!41}a^{5}-\frac{56\!\cdots\!35}{26\!\cdots\!41}a^{4}+\frac{15\!\cdots\!96}{26\!\cdots\!41}a^{3}-\frac{10\!\cdots\!62}{26\!\cdots\!41}a^{2}-\frac{74\!\cdots\!12}{26\!\cdots\!41}a-\frac{42\!\cdots\!95}{26\!\cdots\!41}$, $\frac{83\!\cdots\!17}{26\!\cdots\!41}a^{27}-\frac{13\!\cdots\!32}{26\!\cdots\!41}a^{26}+\frac{13\!\cdots\!93}{26\!\cdots\!41}a^{25}-\frac{33\!\cdots\!12}{26\!\cdots\!41}a^{24}-\frac{73\!\cdots\!78}{26\!\cdots\!41}a^{23}-\frac{78\!\cdots\!47}{26\!\cdots\!41}a^{22}-\frac{81\!\cdots\!24}{26\!\cdots\!41}a^{21}-\frac{54\!\cdots\!71}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!99}{26\!\cdots\!41}a^{19}-\frac{20\!\cdots\!17}{26\!\cdots\!41}a^{18}+\frac{14\!\cdots\!44}{26\!\cdots\!41}a^{17}+\frac{94\!\cdots\!71}{26\!\cdots\!41}a^{16}+\frac{81\!\cdots\!82}{26\!\cdots\!41}a^{15}+\frac{17\!\cdots\!07}{26\!\cdots\!41}a^{14}+\frac{26\!\cdots\!45}{26\!\cdots\!41}a^{13}+\frac{63\!\cdots\!77}{26\!\cdots\!41}a^{12}+\frac{87\!\cdots\!97}{26\!\cdots\!41}a^{11}+\frac{63\!\cdots\!41}{26\!\cdots\!41}a^{10}+\frac{14\!\cdots\!21}{26\!\cdots\!41}a^{9}+\frac{11\!\cdots\!11}{26\!\cdots\!41}a^{8}-\frac{82\!\cdots\!40}{26\!\cdots\!41}a^{7}+\frac{27\!\cdots\!23}{26\!\cdots\!41}a^{6}-\frac{10\!\cdots\!86}{26\!\cdots\!41}a^{5}-\frac{76\!\cdots\!46}{26\!\cdots\!41}a^{4}+\frac{32\!\cdots\!82}{26\!\cdots\!41}a^{3}-\frac{39\!\cdots\!26}{26\!\cdots\!41}a^{2}-\frac{32\!\cdots\!01}{26\!\cdots\!41}a+\frac{65\!\cdots\!25}{26\!\cdots\!41}$, $\frac{92\!\cdots\!78}{26\!\cdots\!41}a^{27}-\frac{10\!\cdots\!88}{26\!\cdots\!41}a^{26}+\frac{15\!\cdots\!17}{26\!\cdots\!41}a^{25}-\frac{20\!\cdots\!88}{26\!\cdots\!41}a^{24}+\frac{12\!\cdots\!25}{26\!\cdots\!41}a^{23}-\frac{83\!\cdots\!34}{26\!\cdots\!41}a^{22}-\frac{76\!\cdots\!32}{26\!\cdots\!41}a^{21}+\frac{20\!\cdots\!76}{26\!\cdots\!41}a^{20}-\frac{22\!\cdots\!76}{26\!\cdots\!41}a^{19}-\frac{10\!\cdots\!87}{26\!\cdots\!41}a^{18}+\frac{14\!\cdots\!91}{26\!\cdots\!41}a^{17}-\frac{77\!\cdots\!89}{26\!\cdots\!41}a^{16}+\frac{86\!\cdots\!99}{26\!\cdots\!41}a^{15}+\frac{11\!\cdots\!82}{26\!\cdots\!41}a^{14}+\frac{21\!\cdots\!55}{26\!\cdots\!41}a^{13}+\frac{55\!\cdots\!47}{26\!\cdots\!41}a^{12}+\frac{79\!\cdots\!35}{26\!\cdots\!41}a^{11}+\frac{45\!\cdots\!18}{26\!\cdots\!41}a^{10}+\frac{23\!\cdots\!20}{26\!\cdots\!41}a^{9}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{8}+\frac{27\!\cdots\!65}{26\!\cdots\!41}a^{7}+\frac{36\!\cdots\!84}{26\!\cdots\!41}a^{6}-\frac{15\!\cdots\!03}{26\!\cdots\!41}a^{5}-\frac{28\!\cdots\!68}{26\!\cdots\!41}a^{4}+\frac{30\!\cdots\!16}{26\!\cdots\!41}a^{3}-\frac{29\!\cdots\!81}{26\!\cdots\!41}a^{2}-\frac{25\!\cdots\!09}{26\!\cdots\!41}a+\frac{15\!\cdots\!62}{26\!\cdots\!41}$, $\frac{46\!\cdots\!10}{26\!\cdots\!41}a^{27}-\frac{19\!\cdots\!77}{26\!\cdots\!41}a^{26}+\frac{85\!\cdots\!56}{26\!\cdots\!41}a^{25}-\frac{63\!\cdots\!93}{26\!\cdots\!41}a^{24}+\frac{13\!\cdots\!16}{26\!\cdots\!41}a^{23}-\frac{71\!\cdots\!29}{26\!\cdots\!41}a^{22}-\frac{40\!\cdots\!19}{26\!\cdots\!41}a^{21}-\frac{27\!\cdots\!76}{26\!\cdots\!41}a^{20}-\frac{16\!\cdots\!17}{26\!\cdots\!41}a^{19}-\frac{44\!\cdots\!78}{26\!\cdots\!41}a^{18}+\frac{56\!\cdots\!68}{26\!\cdots\!41}a^{17}+\frac{38\!\cdots\!61}{26\!\cdots\!41}a^{16}+\frac{55\!\cdots\!47}{26\!\cdots\!41}a^{15}+\frac{75\!\cdots\!09}{26\!\cdots\!41}a^{14}+\frac{21\!\cdots\!99}{26\!\cdots\!41}a^{13}+\frac{34\!\cdots\!12}{26\!\cdots\!41}a^{12}+\frac{61\!\cdots\!96}{26\!\cdots\!41}a^{11}+\frac{50\!\cdots\!88}{26\!\cdots\!41}a^{10}+\frac{12\!\cdots\!79}{26\!\cdots\!41}a^{9}+\frac{35\!\cdots\!19}{26\!\cdots\!41}a^{8}+\frac{95\!\cdots\!95}{26\!\cdots\!41}a^{7}+\frac{46\!\cdots\!80}{26\!\cdots\!41}a^{6}-\frac{51\!\cdots\!86}{26\!\cdots\!41}a^{5}+\frac{30\!\cdots\!67}{26\!\cdots\!41}a^{4}+\frac{10\!\cdots\!21}{26\!\cdots\!41}a^{3}-\frac{19\!\cdots\!75}{26\!\cdots\!41}a^{2}-\frac{12\!\cdots\!33}{26\!\cdots\!41}a-\frac{37\!\cdots\!75}{26\!\cdots\!41}$, $\frac{35\!\cdots\!84}{26\!\cdots\!41}a^{27}+\frac{11\!\cdots\!35}{26\!\cdots\!41}a^{26}+\frac{36\!\cdots\!99}{26\!\cdots\!41}a^{25}+\frac{20\!\cdots\!89}{26\!\cdots\!41}a^{24}-\frac{47\!\cdots\!26}{26\!\cdots\!41}a^{23}+\frac{16\!\cdots\!99}{26\!\cdots\!41}a^{22}-\frac{54\!\cdots\!62}{26\!\cdots\!41}a^{21}-\frac{99\!\cdots\!16}{26\!\cdots\!41}a^{20}-\frac{11\!\cdots\!38}{26\!\cdots\!41}a^{19}-\frac{39\!\cdots\!25}{26\!\cdots\!41}a^{18}+\frac{96\!\cdots\!33}{26\!\cdots\!41}a^{17}+\frac{19\!\cdots\!78}{26\!\cdots\!41}a^{16}+\frac{23\!\cdots\!09}{26\!\cdots\!41}a^{15}+\frac{19\!\cdots\!11}{26\!\cdots\!41}a^{14}+\frac{12\!\cdots\!70}{26\!\cdots\!41}a^{13}+\frac{54\!\cdots\!69}{26\!\cdots\!41}a^{12}+\frac{55\!\cdots\!28}{26\!\cdots\!41}a^{11}+\frac{75\!\cdots\!40}{26\!\cdots\!41}a^{10}+\frac{11\!\cdots\!05}{26\!\cdots\!41}a^{9}+\frac{21\!\cdots\!78}{26\!\cdots\!41}a^{8}-\frac{23\!\cdots\!19}{26\!\cdots\!41}a^{7}+\frac{29\!\cdots\!32}{26\!\cdots\!41}a^{6}-\frac{11\!\cdots\!18}{26\!\cdots\!41}a^{5}-\frac{31\!\cdots\!59}{26\!\cdots\!41}a^{4}+\frac{33\!\cdots\!71}{26\!\cdots\!41}a^{3}-\frac{35\!\cdots\!44}{26\!\cdots\!41}a^{2}-\frac{30\!\cdots\!14}{26\!\cdots\!41}a+\frac{63\!\cdots\!21}{26\!\cdots\!41}$, $\frac{59\!\cdots\!56}{26\!\cdots\!41}a^{27}+\frac{37\!\cdots\!55}{26\!\cdots\!41}a^{26}+\frac{10\!\cdots\!16}{26\!\cdots\!41}a^{25}-\frac{19\!\cdots\!03}{26\!\cdots\!41}a^{24}+\frac{58\!\cdots\!91}{26\!\cdots\!41}a^{23}-\frac{65\!\cdots\!77}{26\!\cdots\!41}a^{22}-\frac{57\!\cdots\!24}{26\!\cdots\!41}a^{21}-\frac{54\!\cdots\!93}{26\!\cdots\!41}a^{20}-\frac{19\!\cdots\!27}{26\!\cdots\!41}a^{19}-\frac{19\!\cdots\!08}{26\!\cdots\!41}a^{18}+\frac{80\!\cdots\!23}{26\!\cdots\!41}a^{17}+\frac{78\!\cdots\!23}{26\!\cdots\!41}a^{16}+\frac{66\!\cdots\!83}{26\!\cdots\!41}a^{15}+\frac{13\!\cdots\!58}{26\!\cdots\!41}a^{14}+\frac{27\!\cdots\!49}{26\!\cdots\!41}a^{13}+\frac{55\!\cdots\!31}{26\!\cdots\!41}a^{12}+\frac{94\!\cdots\!61}{26\!\cdots\!41}a^{11}+\frac{85\!\cdots\!96}{26\!\cdots\!41}a^{10}+\frac{17\!\cdots\!64}{26\!\cdots\!41}a^{9}+\frac{13\!\cdots\!43}{26\!\cdots\!41}a^{8}+\frac{38\!\cdots\!26}{26\!\cdots\!41}a^{7}+\frac{21\!\cdots\!02}{26\!\cdots\!41}a^{6}-\frac{46\!\cdots\!85}{26\!\cdots\!41}a^{5}-\frac{94\!\cdots\!92}{26\!\cdots\!41}a^{4}+\frac{12\!\cdots\!63}{26\!\cdots\!41}a^{3}-\frac{33\!\cdots\!93}{26\!\cdots\!41}a^{2}-\frac{24\!\cdots\!79}{26\!\cdots\!41}a-\frac{51\!\cdots\!13}{26\!\cdots\!41}$ Copy content Toggle raw display (assuming GRH)
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:  \( 34681517373.86067 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 8992}{2\cdot\sqrt{50201655190081835380839261671426578388690948486328125}}\cr\approx \mathstrut & 0.104012020207135 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111)
 
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^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111, 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^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111);
 
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^28 - x^27 + 18*x^26 - 23*x^25 + 29*x^24 - 149*x^23 - 795*x^22 - 8*x^21 - 2927*x^20 + 667*x^19 + 13846*x^18 - 647*x^17 + 109320*x^16 + 98963*x^15 + 313397*x^14 + 494999*x^13 + 789626*x^12 + 291678*x^11 + 1921156*x^10 - 425111*x^9 + 1044768*x^8 + 1156338*x^7 - 1030264*x^6 + 642721*x^5 + 1780368*x^4 - 4481806*x^3 + 897127*x^2 + 1339909*x + 1148111);
 
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

$C_{28}$ (as 28T1):

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 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 7.7.594823321.1, 14.14.27641779937927268828125.1

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 $28$ $28$ R $28$ ${\href{/padicField/11.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ ${\href{/padicField/19.7.0.1}{7} }^{4}$ $28$ R ${\href{/padicField/31.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/41.2.0.1}{2} }^{14}$ $28$ $28$ $28$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

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
\(5\) Copy content Toggle raw display Deg $28$$4$$7$$21$
\(29\) Copy content Toggle raw display 29.14.13.11$x^{14} + 348$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.11$x^{14} + 348$$14$$1$$13$$C_{14}$$[\ ]_{14}$