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Properties

Label 14.0.55029231779...9111.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 17^{7}$
Root discriminant $133.15$
Ramified primes $7, 17$
Class number $35570$ (GRH)
Class group $[35570]$ (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![11183839, -10001649, 11092144, -13116425, 12547871, -6786024, 2490278, -821778, 194649, -36134, 8659, -392, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 168*x^12 - 392*x^11 + 8659*x^10 - 36134*x^9 + 194649*x^8 - 821778*x^7 + 2490278*x^6 - 6786024*x^5 + 12547871*x^4 - 13116425*x^3 + 11092144*x^2 - 10001649*x + 11183839)
 
gp: K = bnfinit(x^14 + 168*x^12 - 392*x^11 + 8659*x^10 - 36134*x^9 + 194649*x^8 - 821778*x^7 + 2490278*x^6 - 6786024*x^5 + 12547871*x^4 - 13116425*x^3 + 11092144*x^2 - 10001649*x + 11183839, 1)
 

Normalized defining polynomial

\( x^{14} + 168 x^{12} - 392 x^{11} + 8659 x^{10} - 36134 x^{9} + 194649 x^{8} - 821778 x^{7} + 2490278 x^{6} - 6786024 x^{5} + 12547871 x^{4} - 13116425 x^{3} + 11092144 x^{2} - 10001649 x + 11183839 \)

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:  \(-550292317794853063315721009111=-\,7^{25}\cdot 17^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.15$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(833=7^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{833}(832,·)$, $\chi_{833}(1,·)$, $\chi_{833}(356,·)$, $\chi_{833}(358,·)$, $\chi_{833}(713,·)$, $\chi_{833}(715,·)$, $\chi_{833}(237,·)$, $\chi_{833}(239,·)$, $\chi_{833}(594,·)$, $\chi_{833}(596,·)$, $\chi_{833}(118,·)$, $\chi_{833}(120,·)$, $\chi_{833}(475,·)$, $\chi_{833}(477,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31} a^{11} - \frac{6}{31} a^{10} - \frac{7}{31} a^{9} - \frac{3}{31} a^{8} + \frac{13}{31} a^{7} - \frac{6}{31} a^{6} - \frac{12}{31} a^{5} - \frac{6}{31} a^{4} + \frac{8}{31} a^{3} + \frac{1}{31} a^{2} + \frac{8}{31} a$, $\frac{1}{31} a^{12} - \frac{12}{31} a^{10} - \frac{14}{31} a^{9} - \frac{5}{31} a^{8} + \frac{10}{31} a^{7} + \frac{14}{31} a^{6} + \frac{15}{31} a^{5} + \frac{3}{31} a^{4} - \frac{13}{31} a^{3} + \frac{14}{31} a^{2} - \frac{14}{31} a$, $\frac{1}{123821780050264737185583080121909825815954915159} a^{13} - \frac{1519462977114846217347852492220613932114876422}{123821780050264737185583080121909825815954915159} a^{12} - \frac{319293807824500388835726697454343538864210021}{123821780050264737185583080121909825815954915159} a^{11} - \frac{1394088542591474204157300246142629615685551654}{3994250969363378618889776778126123413417900489} a^{10} - \frac{12539369267472663703223115352078453126795363376}{123821780050264737185583080121909825815954915159} a^{9} - \frac{21356373944429267274872225449930732248701382760}{123821780050264737185583080121909825815954915159} a^{8} - \frac{24411085497891443574581745778783029555342650789}{123821780050264737185583080121909825815954915159} a^{7} - \frac{59584288907422309086581414056920421169809471752}{123821780050264737185583080121909825815954915159} a^{6} - \frac{39065894428796553141748485481740171781629134591}{123821780050264737185583080121909825815954915159} a^{5} - \frac{11908298752799062890583624846020425951556508421}{123821780050264737185583080121909825815954915159} a^{4} + \frac{13631391972367777820501600371407834425056320474}{123821780050264737185583080121909825815954915159} a^{3} + \frac{1964858047984978156425524950698111878803815314}{3994250969363378618889776778126123413417900489} a^{2} - \frac{5999416264921001701241461351476944837550897360}{123821780050264737185583080121909825815954915159} a + \frac{1365008547114992581242748443620945632020771732}{3994250969363378618889776778126123413417900489}$

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

Class group and class number

$C_{35570}$, which has order $35570$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-119}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$17$17.14.7.2$x^{14} - 24137569 x^{2} + 1231016019$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$