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Properties

Label 14.0.95269346856...3264.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{27}\cdot 3^{12}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}$
Root discriminant $1924.03$
Ramified primes $2, 3, 7, 11, 13$
Class number $504$ (GRH)
Class group $[2, 6, 42]$ (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8031827600, 32480448, 2162410432, -6246240, 249508896, 144144, 15994160, -264, 615160, 0, 14196, 0, 182, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 182*x^12 + 14196*x^10 + 615160*x^8 - 264*x^7 + 15994160*x^6 + 144144*x^5 + 249508896*x^4 - 6246240*x^3 + 2162410432*x^2 + 32480448*x + 8031827600)
 
gp: K = bnfinit(x^14 + 182*x^12 + 14196*x^10 + 615160*x^8 - 264*x^7 + 15994160*x^6 + 144144*x^5 + 249508896*x^4 - 6246240*x^3 + 2162410432*x^2 + 32480448*x + 8031827600, 1)
 

Normalized defining polynomial

\( x^{14} + 182 x^{12} + 14196 x^{10} + 615160 x^{8} - 264 x^{7} + 15994160 x^{6} + 144144 x^{5} + 249508896 x^{4} - 6246240 x^{3} + 2162410432 x^{2} + 32480448 x + 8031827600 \)

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:  \(-9526934685633986238015184265652194712710283264=-\,2^{27}\cdot 3^{12}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1924.03$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{8} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{4450711809606852637542054084707692744264} a^{13} + \frac{273831134849692729835804818650044192203}{4450711809606852637542054084707692744264} a^{12} + \frac{7620644781229469510147943431236260650}{556338976200856579692756760588461593033} a^{11} + \frac{83664646596427391860372171720117368293}{4450711809606852637542054084707692744264} a^{10} + \frac{61641704473567844501072960172089629767}{556338976200856579692756760588461593033} a^{9} + \frac{144653598128579162320626602449332580783}{2225355904803426318771027042353846372132} a^{8} + \frac{74207054652263155376177739864470821615}{2225355904803426318771027042353846372132} a^{7} + \frac{7847491695102036325025045450931865826}{556338976200856579692756760588461593033} a^{6} + \frac{240836115778298834239302765837394131951}{1112677952401713159385513521176923186066} a^{5} + \frac{103261964058338308178370092805493229588}{556338976200856579692756760588461593033} a^{4} - \frac{86535510031606324711355991436878655701}{1112677952401713159385513521176923186066} a^{3} - \frac{148331352590076503780533218558953530495}{556338976200856579692756760588461593033} a^{2} + \frac{200456437557399420131921060045762770196}{556338976200856579692756760588461593033} a + \frac{162046050626215879914037385693637571922}{556338976200856579692756760588461593033}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{42}$, which has order $504$ (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:  \( 1773447701636632.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{-26}) \), 7.1.68069081958026688.23

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.27.21$x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$7$7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$11$11.14.12.1$x^{14} - 11 x^{7} + 847$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$