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Properties

Label 14.0.47448990293...6087.3
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $47383$ (GRH)
Class group $[7, 6769]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1787701529889, -206678117415, 263153797984, -43947685383, 12080530203, -2112867848, 215110980, -38066328, 5169395, -198534, 11571, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 11571*x^10 - 198534*x^9 + 5169395*x^8 - 38066328*x^7 + 215110980*x^6 - 2112867848*x^5 + 12080530203*x^4 - 43947685383*x^3 + 263153797984*x^2 - 206678117415*x + 1787701529889)
 
gp: K = bnfinit(x^14 - 812*x^11 + 11571*x^10 - 198534*x^9 + 5169395*x^8 - 38066328*x^7 + 215110980*x^6 - 2112867848*x^5 + 12080530203*x^4 - 43947685383*x^3 + 263153797984*x^2 - 206678117415*x + 1787701529889, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} + 11571 x^{10} - 198534 x^{9} + 5169395 x^{8} - 38066328 x^{7} + 215110980 x^{6} - 2112867848 x^{5} + 12080530203 x^{4} - 43947685383 x^{3} + 263153797984 x^{2} - 206678117415 x + 1787701529889 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-474489902930063357496193840307474416087=-\,7^{25}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $578.88$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(384,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(517,·)$, $\chi_{1421}(169,·)$, $\chi_{1421}(426,·)$, $\chi_{1421}(141,·)$, $\chi_{1421}(944,·)$, $\chi_{1421}(1009,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(692,·)$, $\chi_{1421}(645,·)$, $\chi_{1421}(951,·)$, $\chi_{1421}(1408,·)$, $\chi_{1421}(1093,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{957} a^{7} - \frac{4}{33} a^{6} + \frac{1}{33} a^{5} + \frac{4}{33} a^{4} + \frac{3}{11} a^{3} + \frac{2}{33} a^{2} + \frac{1}{33} a$, $\frac{1}{2871} a^{8} + \frac{1}{2871} a^{7} + \frac{2}{33} a^{6} - \frac{1}{9} a^{5} + \frac{5}{33} a^{4} - \frac{34}{99} a^{3} - \frac{40}{99} a^{2} + \frac{2}{11} a$, $\frac{1}{2871} a^{9} - \frac{1}{2871} a^{7} - \frac{14}{99} a^{6} - \frac{16}{99} a^{5} + \frac{14}{99} a^{4} + \frac{5}{11} a^{3} - \frac{26}{99} a^{2} + \frac{2}{33} a$, $\frac{1}{83259} a^{10} - \frac{442}{2871} a^{6} + \frac{125}{957} a^{5} + \frac{65}{957} a^{4} - \frac{224}{957} a^{3} + \frac{37}{99} a^{2} - \frac{5}{33} a$, $\frac{1}{749331} a^{11} + \frac{1}{249777} a^{10} + \frac{1}{8613} a^{9} + \frac{4}{25839} a^{8} + \frac{1}{8613} a^{7} + \frac{205}{8613} a^{6} + \frac{178}{2349} a^{5} + \frac{14}{99} a^{4} - \frac{3166}{8613} a^{3} + \frac{119}{891} a^{2} + \frac{64}{297} a - \frac{1}{9}$, $\frac{1}{114020452953} a^{12} - \frac{2471}{3931739757} a^{11} - \frac{4975}{1310579919} a^{10} - \frac{458837}{3931739757} a^{9} + \frac{432383}{3931739757} a^{8} + \frac{16649}{436859973} a^{7} + \frac{201194075}{3931739757} a^{6} + \frac{61994455}{3931739757} a^{5} - \frac{7001999}{45192411} a^{4} - \frac{3297497}{12325203} a^{3} - \frac{213667}{4675077} a^{2} - \frac{33487}{141669} a - \frac{611}{4293}$, $\frac{1}{2230872802850905657661753896248589927306509} a^{13} - \frac{3716318509143022426600106938103}{2230872802850905657661753896248589927306509} a^{12} - \frac{46166499018361099554042339571219120}{76926648374169160609025996422365169907121} a^{11} - \frac{10232372175786378463390798184913706}{6993331670379014600820545129305924537011} a^{10} + \frac{10272211415870076937819648957284815635}{76926648374169160609025996422365169907121} a^{9} + \frac{24792410998150608923732096434630802}{718940639010926734663794359087524952403} a^{8} - \frac{2848000766970388484368684854913511623}{5917434490320704662232768955566551531317} a^{7} + \frac{10075054982788570259774982609845495637776}{76926648374169160609025996422365169907121} a^{6} - \frac{8533035677247751362590749359011289491005}{76926648374169160609025996422365169907121} a^{5} - \frac{174432920153449850272628671793228087908}{2652643047385143469276758497322936893349} a^{4} - \frac{1186386879022404879274412290079726075887}{2652643047385143469276758497322936893349} a^{3} - \frac{13857642213602340921950153536857312044}{91470449909832533423336499907687479081} a^{2} - \frac{900271249257378860199894898496574074}{2771831815449470709798075754778408457} a + \frac{34685395264576854469478332604768666}{83994903498468809387820477417527529}$

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

Class group and class number

$C_{7}\times C_{6769}$, which has order $47383$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4115791439.997621 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.14.25.54$x^{14} - 119 x^{13} + 133 x^{12} - 98 x^{11} + 98 x^{10} + 98 x^{9} - 49 x^{8} + 21 x^{7} + 147 x^{6} + 98 x^{5} - 49 x^{4} + 49 x^{2} - 147 x + 112$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$