Properties

Label 28.0.135...944.2
Degree $28$
Signature $[0, 14]$
Discriminant $1.352\times 10^{53}$
Root discriminant \(78.98\)
Ramified primes $2,3,29$
Class number $2688$ (GRH)
Class group [2, 4, 4, 84] (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761)
 
gp: K = bnfinit(y^28 - 31*y^26 + 354*y^24 - 1731*y^22 + 2216*y^20 + 8564*y^18 - 4122*y^16 - 121607*y^14 + 165323*y^12 - 180077*y^10 + 1311812*y^8 - 3636433*y^6 + 15557681*y^4 + 4370782*y^2 + 2825761, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761)
 

\( x^{28} - 31 x^{26} + 354 x^{24} - 1731 x^{22} + 2216 x^{20} + 8564 x^{18} - 4122 x^{16} - 121607 x^{14} + 165323 x^{12} - 180077 x^{10} + 1311812 x^{8} - 3636433 x^{6} + \cdots + 2825761 \) 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:   \(135171579942192030712001098144632895138136441638354944\) \(\medspace = 2^{28}\cdot 3^{14}\cdot 29^{26}\) 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:  \(78.98\)
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:  $2\cdot 3^{1/2}29^{13/14}\approx 78.98263547298767$
Ramified primes:   \(2\), \(3\), \(29\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(149,·)$, $\chi_{348}(1,·)$, $\chi_{348}(323,·)$, $\chi_{348}(5,·)$, $\chi_{348}(7,·)$, $\chi_{348}(103,·)$, $\chi_{348}(139,·)$, $\chi_{348}(245,·)$, $\chi_{348}(209,·)$, $\chi_{348}(277,·)$, $\chi_{348}(343,·)$, $\chi_{348}(25,·)$, $\chi_{348}(71,·)$, $\chi_{348}(347,·)$, $\chi_{348}(199,·)$, $\chi_{348}(223,·)$, $\chi_{348}(35,·)$, $\chi_{348}(167,·)$, $\chi_{348}(169,·)$, $\chi_{348}(299,·)$, $\chi_{348}(173,·)$, $\chi_{348}(175,·)$, $\chi_{348}(49,·)$, $\chi_{348}(179,·)$, $\chi_{348}(181,·)$, $\chi_{348}(313,·)$, $\chi_{348}(125,·)$, $\chi_{348}(341,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17}a^{12}-\frac{4}{17}a^{8}-\frac{1}{17}a^{4}+\frac{4}{17}$, $\frac{1}{17}a^{13}-\frac{4}{17}a^{9}-\frac{1}{17}a^{5}+\frac{4}{17}a$, $\frac{1}{17}a^{14}-\frac{4}{17}a^{10}-\frac{1}{17}a^{6}+\frac{4}{17}a^{2}$, $\frac{1}{17}a^{15}-\frac{4}{17}a^{11}-\frac{1}{17}a^{7}+\frac{4}{17}a^{3}$, $\frac{1}{17}a^{16}-\frac{1}{17}$, $\frac{1}{17}a^{17}-\frac{1}{17}a$, $\frac{1}{17}a^{18}-\frac{1}{17}a^{2}$, $\frac{1}{17}a^{19}-\frac{1}{17}a^{3}$, $\frac{1}{289}a^{20}+\frac{3}{289}a^{18}+\frac{5}{289}a^{16}-\frac{7}{289}a^{14}-\frac{4}{289}a^{12}+\frac{79}{289}a^{10}+\frac{101}{289}a^{8}+\frac{143}{289}a^{6}-\frac{99}{289}a^{4}-\frac{133}{289}a^{2}-\frac{89}{289}$, $\frac{1}{289}a^{21}+\frac{3}{289}a^{19}+\frac{5}{289}a^{17}-\frac{7}{289}a^{15}-\frac{4}{289}a^{13}+\frac{79}{289}a^{11}+\frac{101}{289}a^{9}+\frac{143}{289}a^{7}-\frac{99}{289}a^{5}-\frac{133}{289}a^{3}-\frac{89}{289}a$, $\frac{1}{289}a^{22}-\frac{4}{289}a^{18}-\frac{5}{289}a^{16}+\frac{6}{289}a^{12}-\frac{4}{17}a^{10}-\frac{109}{289}a^{8}+\frac{67}{289}a^{6}-\frac{40}{289}a^{4}-\frac{47}{289}a^{2}-\frac{90}{289}$, $\frac{1}{11849}a^{23}+\frac{3}{11849}a^{21}-\frac{46}{11849}a^{19}+\frac{78}{11849}a^{17}+\frac{285}{11849}a^{15}-\frac{74}{11849}a^{13}-\frac{5101}{11849}a^{11}-\frac{2135}{11849}a^{9}+\frac{1924}{11849}a^{7}+\frac{3777}{11849}a^{5}-\frac{4662}{11849}a^{3}+\frac{44}{697}a$, $\frac{1}{93263479}a^{24}-\frac{133124}{93263479}a^{22}-\frac{115871}{93263479}a^{20}+\frac{1503138}{93263479}a^{18}-\frac{466623}{93263479}a^{16}-\frac{1002196}{93263479}a^{14}+\frac{2359615}{93263479}a^{12}-\frac{9827703}{93263479}a^{10}+\frac{8614179}{93263479}a^{8}-\frac{44994010}{93263479}a^{6}-\frac{18284553}{93263479}a^{4}+\frac{2405029}{93263479}a^{2}+\frac{1008146}{2274719}$, $\frac{1}{93263479}a^{25}+\frac{683}{93263479}a^{23}-\frac{37161}{93263479}a^{21}-\frac{134030}{93263479}a^{19}-\frac{2615406}{93263479}a^{17}+\frac{989167}{93263479}a^{15}-\frac{765172}{93263479}a^{13}-\frac{4680069}{93263479}a^{11}+\frac{41451991}{93263479}a^{9}+\frac{18178636}{93263479}a^{7}-\frac{46014086}{93263479}a^{5}+\frac{8528667}{93263479}a^{3}+\frac{16532465}{93263479}a$, $\frac{1}{24\!\cdots\!53}a^{26}+\frac{88\!\cdots\!11}{24\!\cdots\!53}a^{24}-\frac{35\!\cdots\!84}{24\!\cdots\!53}a^{22}+\frac{42\!\cdots\!89}{24\!\cdots\!53}a^{20}+\frac{12\!\cdots\!64}{24\!\cdots\!53}a^{18}-\frac{38\!\cdots\!10}{24\!\cdots\!53}a^{16}+\frac{29\!\cdots\!91}{24\!\cdots\!53}a^{14}-\frac{65\!\cdots\!68}{24\!\cdots\!53}a^{12}-\frac{11\!\cdots\!79}{24\!\cdots\!53}a^{10}+\frac{58\!\cdots\!70}{24\!\cdots\!53}a^{8}+\frac{96\!\cdots\!20}{24\!\cdots\!53}a^{6}-\frac{18\!\cdots\!95}{24\!\cdots\!53}a^{4}-\frac{51\!\cdots\!21}{24\!\cdots\!53}a^{2}-\frac{11\!\cdots\!88}{60\!\cdots\!33}$, $\frac{1}{10\!\cdots\!73}a^{27}-\frac{52\!\cdots\!29}{10\!\cdots\!73}a^{25}-\frac{41\!\cdots\!01}{10\!\cdots\!73}a^{23}+\frac{40\!\cdots\!92}{10\!\cdots\!73}a^{21}+\frac{29\!\cdots\!59}{10\!\cdots\!73}a^{19}+\frac{10\!\cdots\!53}{10\!\cdots\!73}a^{17}-\frac{27\!\cdots\!38}{10\!\cdots\!73}a^{15}-\frac{27\!\cdots\!15}{10\!\cdots\!73}a^{13}+\frac{21\!\cdots\!60}{10\!\cdots\!73}a^{11}-\frac{70\!\cdots\!65}{10\!\cdots\!73}a^{9}+\frac{39\!\cdots\!54}{10\!\cdots\!73}a^{7}+\frac{50\!\cdots\!95}{10\!\cdots\!73}a^{5}+\frac{17\!\cdots\!98}{10\!\cdots\!73}a^{3}+\frac{17\!\cdots\!03}{10\!\cdots\!73}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $17$

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{84}$, which has order $2688$ (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:   \( \frac{2729773008375457211}{17614789467148212613700969} a^{27} - \frac{84186284058644055375}{17614789467148212613700969} a^{25} + \frac{949112912447047986756}{17614789467148212613700969} a^{23} - \frac{4466117280551868347775}{17614789467148212613700969} a^{21} + \frac{4256287364530340094435}{17614789467148212613700969} a^{19} + \frac{28175401974806682070614}{17614789467148212613700969} a^{17} - \frac{6386052889700357177231}{17614789467148212613700969} a^{15} - \frac{369537423585515603088009}{17614789467148212613700969} a^{13} + \frac{382076733259297499664894}{17614789467148212613700969} a^{11} - \frac{259981068865718655103}{38044901656907586638663} a^{9} + \frac{3362786515297414681391643}{17614789467148212613700969} a^{7} - \frac{8483773797085126506553065}{17614789467148212613700969} a^{5} + \frac{37753029883558878099525217}{17614789467148212613700969} a^{3} + \frac{27180174763845314591565123}{17614789467148212613700969} a \)  (order $4$) 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:   $\frac{24\!\cdots\!90}{14\!\cdots\!09}a^{26}-\frac{11\!\cdots\!09}{24\!\cdots\!53}a^{24}+\frac{98\!\cdots\!95}{24\!\cdots\!53}a^{22}-\frac{11\!\cdots\!98}{24\!\cdots\!53}a^{20}-\frac{24\!\cdots\!72}{24\!\cdots\!53}a^{18}+\frac{11\!\cdots\!49}{24\!\cdots\!53}a^{16}-\frac{43\!\cdots\!10}{24\!\cdots\!53}a^{14}-\frac{15\!\cdots\!55}{24\!\cdots\!53}a^{12}-\frac{18\!\cdots\!63}{24\!\cdots\!53}a^{10}+\frac{24\!\cdots\!52}{24\!\cdots\!53}a^{8}+\frac{27\!\cdots\!26}{24\!\cdots\!53}a^{6}+\frac{41\!\cdots\!88}{24\!\cdots\!53}a^{4}+\frac{13\!\cdots\!87}{24\!\cdots\!53}a^{2}+\frac{41\!\cdots\!17}{60\!\cdots\!33}$, $\frac{60\!\cdots\!01}{24\!\cdots\!53}a^{26}-\frac{17\!\cdots\!01}{24\!\cdots\!53}a^{24}+\frac{18\!\cdots\!63}{24\!\cdots\!53}a^{22}-\frac{59\!\cdots\!65}{24\!\cdots\!53}a^{20}-\frac{14\!\cdots\!72}{24\!\cdots\!53}a^{18}+\frac{10\!\cdots\!34}{24\!\cdots\!53}a^{16}+\frac{15\!\cdots\!71}{24\!\cdots\!53}a^{14}-\frac{57\!\cdots\!96}{14\!\cdots\!09}a^{12}-\frac{25\!\cdots\!10}{24\!\cdots\!53}a^{10}+\frac{29\!\cdots\!90}{24\!\cdots\!53}a^{8}+\frac{28\!\cdots\!03}{24\!\cdots\!53}a^{6}+\frac{28\!\cdots\!50}{24\!\cdots\!53}a^{4}+\frac{62\!\cdots\!37}{14\!\cdots\!09}a^{2}+\frac{11\!\cdots\!74}{60\!\cdots\!33}$, $\frac{85\!\cdots\!62}{24\!\cdots\!53}a^{26}-\frac{23\!\cdots\!48}{24\!\cdots\!53}a^{24}+\frac{20\!\cdots\!07}{24\!\cdots\!53}a^{22}-\frac{29\!\cdots\!16}{24\!\cdots\!53}a^{20}-\frac{40\!\cdots\!30}{24\!\cdots\!53}a^{18}+\frac{13\!\cdots\!11}{24\!\cdots\!53}a^{16}+\frac{43\!\cdots\!59}{24\!\cdots\!53}a^{14}-\frac{20\!\cdots\!50}{24\!\cdots\!53}a^{12}-\frac{40\!\cdots\!70}{53\!\cdots\!31}a^{10}+\frac{68\!\cdots\!95}{24\!\cdots\!53}a^{8}+\frac{27\!\cdots\!68}{24\!\cdots\!53}a^{6}+\frac{28\!\cdots\!37}{24\!\cdots\!53}a^{4}+\frac{82\!\cdots\!89}{24\!\cdots\!53}a^{2}+\frac{65\!\cdots\!78}{60\!\cdots\!33}$, $\frac{94\!\cdots\!29}{24\!\cdots\!53}a^{26}-\frac{25\!\cdots\!54}{24\!\cdots\!53}a^{24}+\frac{12\!\cdots\!95}{14\!\cdots\!09}a^{22}-\frac{23\!\cdots\!78}{24\!\cdots\!53}a^{20}-\frac{46\!\cdots\!85}{24\!\cdots\!53}a^{18}+\frac{42\!\cdots\!70}{60\!\cdots\!33}a^{16}+\frac{19\!\cdots\!66}{24\!\cdots\!53}a^{14}-\frac{11\!\cdots\!68}{24\!\cdots\!53}a^{12}-\frac{22\!\cdots\!83}{24\!\cdots\!53}a^{10}+\frac{47\!\cdots\!60}{24\!\cdots\!53}a^{8}-\frac{66\!\cdots\!28}{24\!\cdots\!53}a^{6}-\frac{21\!\cdots\!53}{24\!\cdots\!53}a^{4}-\frac{15\!\cdots\!76}{24\!\cdots\!53}a^{2}+\frac{10\!\cdots\!36}{60\!\cdots\!33}$, $\frac{71\!\cdots\!26}{24\!\cdots\!53}a^{26}-\frac{18\!\cdots\!67}{24\!\cdots\!53}a^{24}+\frac{13\!\cdots\!09}{24\!\cdots\!53}a^{22}+\frac{13\!\cdots\!51}{24\!\cdots\!53}a^{20}-\frac{84\!\cdots\!90}{35\!\cdots\!49}a^{18}+\frac{20\!\cdots\!23}{24\!\cdots\!53}a^{16}+\frac{13\!\cdots\!06}{24\!\cdots\!53}a^{14}-\frac{11\!\cdots\!25}{24\!\cdots\!53}a^{12}-\frac{27\!\cdots\!29}{24\!\cdots\!53}a^{10}+\frac{85\!\cdots\!00}{24\!\cdots\!53}a^{8}-\frac{78\!\cdots\!19}{24\!\cdots\!53}a^{6}+\frac{20\!\cdots\!34}{24\!\cdots\!53}a^{4}+\frac{48\!\cdots\!05}{24\!\cdots\!53}a^{2}+\frac{20\!\cdots\!28}{60\!\cdots\!33}$, $\frac{12\!\cdots\!92}{24\!\cdots\!53}a^{26}-\frac{34\!\cdots\!57}{24\!\cdots\!53}a^{24}+\frac{29\!\cdots\!02}{24\!\cdots\!53}a^{22}-\frac{41\!\cdots\!14}{24\!\cdots\!53}a^{20}-\frac{65\!\cdots\!02}{24\!\cdots\!53}a^{18}+\frac{25\!\cdots\!60}{24\!\cdots\!53}a^{16}+\frac{39\!\cdots\!49}{24\!\cdots\!53}a^{14}-\frac{22\!\cdots\!05}{24\!\cdots\!53}a^{12}-\frac{37\!\cdots\!73}{24\!\cdots\!53}a^{10}+\frac{93\!\cdots\!47}{24\!\cdots\!53}a^{8}+\frac{54\!\cdots\!94}{24\!\cdots\!53}a^{6}+\frac{28\!\cdots\!25}{24\!\cdots\!53}a^{4}+\frac{84\!\cdots\!76}{24\!\cdots\!53}a^{2}+\frac{16\!\cdots\!28}{60\!\cdots\!33}$, $\frac{11\!\cdots\!78}{10\!\cdots\!73}a^{27}-\frac{71\!\cdots\!26}{24\!\cdots\!53}a^{26}-\frac{37\!\cdots\!94}{10\!\cdots\!73}a^{25}+\frac{18\!\cdots\!67}{24\!\cdots\!53}a^{24}+\frac{43\!\cdots\!67}{10\!\cdots\!73}a^{23}-\frac{13\!\cdots\!09}{24\!\cdots\!53}a^{22}-\frac{22\!\cdots\!54}{10\!\cdots\!73}a^{21}-\frac{13\!\cdots\!51}{24\!\cdots\!53}a^{20}+\frac{35\!\cdots\!12}{10\!\cdots\!73}a^{19}+\frac{84\!\cdots\!90}{35\!\cdots\!49}a^{18}+\frac{90\!\cdots\!57}{10\!\cdots\!73}a^{17}-\frac{20\!\cdots\!23}{24\!\cdots\!53}a^{16}-\frac{96\!\cdots\!13}{10\!\cdots\!73}a^{15}-\frac{13\!\cdots\!06}{24\!\cdots\!53}a^{14}-\frac{14\!\cdots\!57}{10\!\cdots\!73}a^{13}+\frac{11\!\cdots\!25}{24\!\cdots\!53}a^{12}+\frac{25\!\cdots\!37}{10\!\cdots\!73}a^{11}+\frac{27\!\cdots\!29}{24\!\cdots\!53}a^{10}-\frac{31\!\cdots\!30}{10\!\cdots\!73}a^{9}-\frac{85\!\cdots\!00}{24\!\cdots\!53}a^{8}+\frac{99\!\cdots\!80}{59\!\cdots\!69}a^{7}+\frac{78\!\cdots\!19}{24\!\cdots\!53}a^{6}-\frac{47\!\cdots\!58}{10\!\cdots\!73}a^{5}-\frac{20\!\cdots\!34}{24\!\cdots\!53}a^{4}+\frac{20\!\cdots\!53}{10\!\cdots\!73}a^{3}-\frac{48\!\cdots\!05}{24\!\cdots\!53}a^{2}-\frac{32\!\cdots\!62}{10\!\cdots\!73}a-\frac{38\!\cdots\!27}{60\!\cdots\!33}$, $\frac{38\!\cdots\!32}{59\!\cdots\!69}a^{27}+\frac{30\!\cdots\!41}{43\!\cdots\!31}a^{26}-\frac{20\!\cdots\!55}{10\!\cdots\!73}a^{25}-\frac{95\!\cdots\!79}{43\!\cdots\!31}a^{24}+\frac{22\!\cdots\!50}{10\!\cdots\!73}a^{23}+\frac{10\!\cdots\!31}{43\!\cdots\!31}a^{22}-\frac{10\!\cdots\!27}{10\!\cdots\!73}a^{21}-\frac{53\!\cdots\!42}{43\!\cdots\!31}a^{20}+\frac{11\!\cdots\!48}{10\!\cdots\!73}a^{19}+\frac{67\!\cdots\!49}{43\!\cdots\!31}a^{18}+\frac{56\!\cdots\!38}{10\!\cdots\!73}a^{17}+\frac{28\!\cdots\!84}{43\!\cdots\!31}a^{16}+\frac{57\!\cdots\!42}{10\!\cdots\!73}a^{15}-\frac{18\!\cdots\!25}{43\!\cdots\!31}a^{14}-\frac{84\!\cdots\!71}{10\!\cdots\!73}a^{13}-\frac{41\!\cdots\!67}{43\!\cdots\!31}a^{12}+\frac{95\!\cdots\!14}{10\!\cdots\!73}a^{11}+\frac{68\!\cdots\!22}{43\!\cdots\!31}a^{10}-\frac{10\!\cdots\!09}{10\!\cdots\!73}a^{9}-\frac{74\!\cdots\!41}{43\!\cdots\!31}a^{8}+\frac{82\!\cdots\!45}{10\!\cdots\!73}a^{7}+\frac{11\!\cdots\!28}{10\!\cdots\!91}a^{6}-\frac{21\!\cdots\!87}{10\!\cdots\!73}a^{5}-\frac{13\!\cdots\!56}{43\!\cdots\!31}a^{4}+\frac{93\!\cdots\!39}{10\!\cdots\!73}a^{3}+\frac{49\!\cdots\!54}{43\!\cdots\!31}a^{2}+\frac{67\!\cdots\!44}{10\!\cdots\!73}a+\frac{13\!\cdots\!28}{10\!\cdots\!91}$, $\frac{48\!\cdots\!88}{10\!\cdots\!73}a^{27}+\frac{97\!\cdots\!51}{24\!\cdots\!53}a^{26}-\frac{18\!\cdots\!50}{10\!\cdots\!73}a^{25}-\frac{27\!\cdots\!08}{24\!\cdots\!53}a^{24}+\frac{15\!\cdots\!35}{59\!\cdots\!69}a^{23}+\frac{27\!\cdots\!59}{24\!\cdots\!53}a^{22}-\frac{16\!\cdots\!28}{10\!\cdots\!73}a^{21}-\frac{10\!\cdots\!65}{24\!\cdots\!53}a^{20}+\frac{27\!\cdots\!65}{10\!\cdots\!73}a^{19}+\frac{66\!\cdots\!43}{24\!\cdots\!53}a^{18}+\frac{80\!\cdots\!19}{10\!\cdots\!73}a^{17}+\frac{30\!\cdots\!77}{24\!\cdots\!53}a^{16}+\frac{15\!\cdots\!94}{10\!\cdots\!73}a^{15}+\frac{12\!\cdots\!46}{24\!\cdots\!53}a^{14}-\frac{27\!\cdots\!29}{10\!\cdots\!73}a^{13}-\frac{28\!\cdots\!57}{24\!\cdots\!53}a^{12}+\frac{17\!\cdots\!82}{10\!\cdots\!73}a^{11}-\frac{48\!\cdots\!05}{24\!\cdots\!53}a^{10}+\frac{96\!\cdots\!86}{10\!\cdots\!73}a^{9}-\frac{23\!\cdots\!64}{85\!\cdots\!77}a^{8}+\frac{42\!\cdots\!11}{10\!\cdots\!73}a^{7}+\frac{78\!\cdots\!17}{24\!\cdots\!53}a^{6}-\frac{67\!\cdots\!75}{10\!\cdots\!73}a^{5}-\frac{51\!\cdots\!65}{24\!\cdots\!53}a^{4}+\frac{56\!\cdots\!70}{10\!\cdots\!73}a^{3}-\frac{14\!\cdots\!67}{24\!\cdots\!53}a^{2}-\frac{77\!\cdots\!95}{10\!\cdots\!73}a+\frac{37\!\cdots\!52}{60\!\cdots\!33}$, $\frac{40\!\cdots\!35}{59\!\cdots\!69}a^{27}+\frac{16\!\cdots\!57}{24\!\cdots\!53}a^{26}-\frac{21\!\cdots\!91}{10\!\cdots\!73}a^{25}-\frac{50\!\cdots\!85}{24\!\cdots\!53}a^{24}+\frac{23\!\cdots\!31}{10\!\cdots\!73}a^{23}+\frac{59\!\cdots\!59}{24\!\cdots\!53}a^{22}-\frac{11\!\cdots\!30}{10\!\cdots\!73}a^{21}-\frac{29\!\cdots\!67}{24\!\cdots\!53}a^{20}+\frac{10\!\cdots\!24}{10\!\cdots\!73}a^{19}+\frac{42\!\cdots\!36}{24\!\cdots\!53}a^{18}+\frac{70\!\cdots\!44}{10\!\cdots\!73}a^{17}+\frac{14\!\cdots\!76}{24\!\cdots\!53}a^{16}-\frac{15\!\cdots\!54}{10\!\cdots\!73}a^{15}-\frac{15\!\cdots\!53}{24\!\cdots\!53}a^{14}-\frac{90\!\cdots\!10}{10\!\cdots\!73}a^{13}-\frac{19\!\cdots\!30}{24\!\cdots\!53}a^{12}+\frac{96\!\cdots\!40}{10\!\cdots\!73}a^{11}+\frac{34\!\cdots\!64}{24\!\cdots\!53}a^{10}-\frac{48\!\cdots\!09}{10\!\cdots\!73}a^{9}-\frac{39\!\cdots\!10}{24\!\cdots\!53}a^{8}+\frac{75\!\cdots\!24}{10\!\cdots\!73}a^{7}+\frac{11\!\cdots\!05}{14\!\cdots\!09}a^{6}-\frac{20\!\cdots\!74}{10\!\cdots\!73}a^{5}-\frac{53\!\cdots\!68}{24\!\cdots\!53}a^{4}+\frac{97\!\cdots\!60}{10\!\cdots\!73}a^{3}+\frac{25\!\cdots\!11}{24\!\cdots\!53}a^{2}+\frac{69\!\cdots\!18}{10\!\cdots\!73}a+\frac{75\!\cdots\!74}{60\!\cdots\!33}$, $\frac{39\!\cdots\!20}{10\!\cdots\!73}a^{27}+\frac{18\!\cdots\!48}{14\!\cdots\!09}a^{26}-\frac{12\!\cdots\!89}{10\!\cdots\!73}a^{25}-\frac{84\!\cdots\!34}{24\!\cdots\!53}a^{24}+\frac{15\!\cdots\!85}{10\!\cdots\!73}a^{23}+\frac{71\!\cdots\!64}{24\!\cdots\!53}a^{22}-\frac{79\!\cdots\!38}{10\!\cdots\!73}a^{21}-\frac{11\!\cdots\!23}{24\!\cdots\!53}a^{20}+\frac{13\!\cdots\!30}{10\!\cdots\!73}a^{19}-\frac{11\!\cdots\!43}{24\!\cdots\!53}a^{18}+\frac{31\!\cdots\!40}{10\!\cdots\!73}a^{17}+\frac{45\!\cdots\!69}{24\!\cdots\!53}a^{16}-\frac{38\!\cdots\!15}{10\!\cdots\!73}a^{15}+\frac{54\!\cdots\!38}{24\!\cdots\!53}a^{14}-\frac{53\!\cdots\!34}{10\!\cdots\!73}a^{13}-\frac{28\!\cdots\!74}{24\!\cdots\!53}a^{12}+\frac{10\!\cdots\!20}{10\!\cdots\!73}a^{11}-\frac{49\!\cdots\!96}{24\!\cdots\!53}a^{10}-\frac{80\!\cdots\!25}{10\!\cdots\!73}a^{9}+\frac{52\!\cdots\!19}{24\!\cdots\!53}a^{8}+\frac{66\!\cdots\!11}{10\!\cdots\!73}a^{7}-\frac{34\!\cdots\!87}{24\!\cdots\!53}a^{6}-\frac{18\!\cdots\!58}{10\!\cdots\!73}a^{5}-\frac{78\!\cdots\!44}{24\!\cdots\!53}a^{4}+\frac{83\!\cdots\!39}{10\!\cdots\!73}a^{3}-\frac{27\!\cdots\!18}{24\!\cdots\!53}a^{2}-\frac{12\!\cdots\!21}{10\!\cdots\!73}a+\frac{13\!\cdots\!45}{60\!\cdots\!33}$, $\frac{15\!\cdots\!96}{10\!\cdots\!73}a^{27}+\frac{57\!\cdots\!27}{24\!\cdots\!53}a^{26}-\frac{44\!\cdots\!22}{10\!\cdots\!73}a^{25}-\frac{97\!\cdots\!47}{14\!\cdots\!09}a^{24}+\frac{44\!\cdots\!45}{10\!\cdots\!73}a^{23}+\frac{16\!\cdots\!86}{24\!\cdots\!53}a^{22}-\frac{14\!\cdots\!11}{10\!\cdots\!73}a^{21}-\frac{54\!\cdots\!49}{24\!\cdots\!53}a^{20}-\frac{15\!\cdots\!74}{10\!\cdots\!73}a^{19}-\frac{10\!\cdots\!10}{24\!\cdots\!53}a^{18}+\frac{13\!\cdots\!00}{10\!\cdots\!73}a^{17}+\frac{17\!\cdots\!87}{60\!\cdots\!33}a^{16}+\frac{42\!\cdots\!31}{10\!\cdots\!73}a^{15}+\frac{82\!\cdots\!08}{14\!\cdots\!09}a^{14}-\frac{15\!\cdots\!65}{10\!\cdots\!73}a^{13}-\frac{75\!\cdots\!14}{24\!\cdots\!53}a^{12}-\frac{35\!\cdots\!64}{10\!\cdots\!73}a^{11}-\frac{16\!\cdots\!74}{24\!\cdots\!53}a^{10}-\frac{64\!\cdots\!26}{10\!\cdots\!73}a^{9}+\frac{95\!\cdots\!03}{24\!\cdots\!53}a^{8}+\frac{41\!\cdots\!04}{10\!\cdots\!73}a^{7}+\frac{17\!\cdots\!83}{24\!\cdots\!53}a^{6}+\frac{45\!\cdots\!31}{10\!\cdots\!73}a^{5}+\frac{84\!\cdots\!17}{24\!\cdots\!53}a^{4}+\frac{16\!\cdots\!14}{10\!\cdots\!73}a^{3}+\frac{38\!\cdots\!12}{24\!\cdots\!53}a^{2}-\frac{10\!\cdots\!55}{10\!\cdots\!73}a+\frac{30\!\cdots\!02}{60\!\cdots\!33}$, $\frac{58\!\cdots\!16}{10\!\cdots\!73}a^{27}+\frac{18\!\cdots\!14}{24\!\cdots\!53}a^{26}-\frac{18\!\cdots\!69}{10\!\cdots\!73}a^{25}-\frac{52\!\cdots\!75}{24\!\cdots\!53}a^{24}+\frac{12\!\cdots\!31}{59\!\cdots\!69}a^{23}+\frac{53\!\cdots\!33}{24\!\cdots\!53}a^{22}-\frac{99\!\cdots\!60}{10\!\cdots\!73}a^{21}-\frac{12\!\cdots\!21}{14\!\cdots\!09}a^{20}+\frac{41\!\cdots\!58}{10\!\cdots\!73}a^{19}+\frac{19\!\cdots\!38}{24\!\cdots\!53}a^{18}+\frac{97\!\cdots\!30}{10\!\cdots\!73}a^{17}+\frac{31\!\cdots\!11}{24\!\cdots\!53}a^{16}-\frac{24\!\cdots\!66}{10\!\cdots\!73}a^{15}+\frac{47\!\cdots\!50}{24\!\cdots\!53}a^{14}-\frac{13\!\cdots\!60}{10\!\cdots\!73}a^{13}-\frac{14\!\cdots\!41}{24\!\cdots\!53}a^{12}+\frac{12\!\cdots\!93}{10\!\cdots\!73}a^{11}-\frac{93\!\cdots\!89}{24\!\cdots\!53}a^{10}+\frac{29\!\cdots\!91}{10\!\cdots\!73}a^{9}-\frac{10\!\cdots\!32}{24\!\cdots\!53}a^{8}+\frac{68\!\cdots\!99}{10\!\cdots\!73}a^{7}+\frac{30\!\cdots\!13}{24\!\cdots\!53}a^{6}-\frac{12\!\cdots\!07}{10\!\cdots\!73}a^{5}-\frac{64\!\cdots\!10}{24\!\cdots\!53}a^{4}+\frac{53\!\cdots\!89}{10\!\cdots\!73}a^{3}+\frac{47\!\cdots\!43}{24\!\cdots\!53}a^{2}+\frac{38\!\cdots\!84}{10\!\cdots\!73}a+\frac{15\!\cdots\!04}{60\!\cdots\!33}$ 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:  \( 4155697700585.085 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 4155697700585.085 \cdot 2688}{4\cdot\sqrt{135171579942192030712001098144632895138136441638354944}}\cr\approx \mathstrut & 1.13524260883005 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761)
 
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 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761, 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 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761);
 
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 - 31*x^26 + 354*x^24 - 1731*x^22 + 2216*x^20 + 8564*x^18 - 4122*x^16 - 121607*x^14 + 165323*x^12 - 180077*x^10 + 1311812*x^8 - 3636433*x^6 + 15557681*x^4 + 4370782*x^2 + 2825761);
 
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

$C_2\times C_{14}$ (as 28T2):

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 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{87}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{87})\), 7.7.594823321.1, 14.14.367656878002019745584627712.1, 14.0.22439994995240462987343.1, 14.0.5796901408038404767744.1

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 R R ${\href{/padicField/5.14.0.1}{14} }^{2}$ ${\href{/padicField/7.14.0.1}{14} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.1.0.1}{1} }^{28}$ ${\href{/padicField/19.14.0.1}{14} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R ${\href{/padicField/31.14.0.1}{14} }^{2}$ ${\href{/padicField/37.14.0.1}{14} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{28}$ ${\href{/padicField/43.14.0.1}{14} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

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
\(2\) Copy content Toggle raw display 2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
\(3\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(29\) 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}$