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Properties

Label 14.0.17958157356...8224.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 13^{12}\cdot 19^{6}$
Root discriminant $63.66$
Ramified primes $2, 13, 19$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_7^2$ (as 14T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47545, -114686, 151847, -122460, 71164, -39026, 18996, -7046, 3072, -832, 232, -78, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 15*x^12 - 78*x^11 + 232*x^10 - 832*x^9 + 3072*x^8 - 7046*x^7 + 18996*x^6 - 39026*x^5 + 71164*x^4 - 122460*x^3 + 151847*x^2 - 114686*x + 47545)
 
gp: K = bnfinit(x^14 + 15*x^12 - 78*x^11 + 232*x^10 - 832*x^9 + 3072*x^8 - 7046*x^7 + 18996*x^6 - 39026*x^5 + 71164*x^4 - 122460*x^3 + 151847*x^2 - 114686*x + 47545, 1)
 

Normalized defining polynomial

\( x^{14} + 15 x^{12} - 78 x^{11} + 232 x^{10} - 832 x^{9} + 3072 x^{8} - 7046 x^{7} + 18996 x^{6} - 39026 x^{5} + 71164 x^{4} - 122460 x^{3} + 151847 x^{2} - 114686 x + 47545 \)

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:  \(-17958157356238627647668224=-\,2^{14}\cdot 13^{12}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.66$
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:  $2, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{10} - \frac{1}{20} a^{9} - \frac{1}{4} a^{7} + \frac{1}{10} a^{6} - \frac{7}{20} a^{5} - \frac{1}{10} a^{4} + \frac{7}{20} a^{3} + \frac{2}{5} a^{2} - \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{44080} a^{12} - \frac{83}{5510} a^{11} + \frac{11}{760} a^{10} - \frac{1119}{22040} a^{9} + \frac{419}{4408} a^{8} - \frac{3119}{22040} a^{7} + \frac{2857}{22040} a^{6} - \frac{677}{11020} a^{5} + \frac{6323}{22040} a^{4} + \frac{5417}{22040} a^{3} + \frac{1569}{4408} a^{2} - \frac{7681}{22040} a - \frac{591}{8816}$, $\frac{1}{1987535776458246068000} a^{13} + \frac{14745846443418917}{1987535776458246068000} a^{12} - \frac{19884655870594238323}{993767888229123034000} a^{11} + \frac{594264971199369123}{5230357306469068600} a^{10} + \frac{4465041566888126239}{248441972057280758500} a^{9} + \frac{58714039780327445843}{496883944114561517000} a^{8} - \frac{22781164214019466163}{248441972057280758500} a^{7} + \frac{183556339971610262043}{993767888229123034000} a^{6} - \frac{492802200471494545421}{993767888229123034000} a^{5} + \frac{2021719697833036027}{5230357306469068600} a^{4} + \frac{195855326656815122071}{496883944114561517000} a^{3} - \frac{54056714318333038183}{496883944114561517000} a^{2} + \frac{508168173972209359103}{1987535776458246068000} a - \frac{389650517773867471}{10743436629504032800}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( \frac{46364270629}{5240700794880016} a^{13} + \frac{443506997245}{5240700794880016} a^{12} + \frac{648587239079}{2620350397440008} a^{11} + \frac{207640899336}{327543799680001} a^{10} - \frac{855707296600}{327543799680001} a^{9} + \frac{1353115476336}{327543799680001} a^{8} - \frac{24128243517651}{1310175198720004} a^{7} + \frac{265331901764527}{2620350397440008} a^{6} - \frac{213035671775979}{2620350397440008} a^{5} + \frac{348400506737829}{655087599360002} a^{4} - \frac{760872012214769}{1310175198720004} a^{3} + \frac{723275753131971}{655087599360002} a^{2} - \frac{9912717585618913}{5240700794880016} a + \frac{6317672742680273}{5240700794880016} \) (order $4$)
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:  \( 27147114.557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7^2$ (as 14T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 25 conjugacy class representatives for $D_7^2$
Character table for $D_7^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$