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Properties

Label 14.0.13410686196...0807.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}$
Root discriminant $32.29$
Ramified prime $7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7 \wr C_2$ (as 14T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, 133, 644, 791, 1953, 1176, 1176, 284, 189, -70, -21, -28, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^11 - 21*x^10 - 70*x^9 + 189*x^8 + 284*x^7 + 1176*x^6 + 1176*x^5 + 1953*x^4 + 791*x^3 + 644*x^2 + 133*x + 53)
 
gp: K = bnfinit(x^14 - 28*x^11 - 21*x^10 - 70*x^9 + 189*x^8 + 284*x^7 + 1176*x^6 + 1176*x^5 + 1953*x^4 + 791*x^3 + 644*x^2 + 133*x + 53, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{11} - 21 x^{10} - 70 x^{9} + 189 x^{8} + 284 x^{7} + 1176 x^{6} + 1176 x^{5} + 1953 x^{4} + 791 x^{3} + 644 x^{2} + 133 x + 53 \)

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:  \(-1341068619663964900807=-\,7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.29$
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:  $7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{589} a^{12} + \frac{64}{589} a^{11} - \frac{129}{589} a^{10} - \frac{87}{589} a^{9} - \frac{88}{589} a^{8} + \frac{227}{589} a^{7} + \frac{7}{31} a^{6} - \frac{222}{589} a^{5} - \frac{3}{19} a^{4} + \frac{198}{589} a^{3} - \frac{2}{31} a^{2} - \frac{44}{589} a - \frac{63}{589}$, $\frac{1}{48256443282289} a^{13} + \frac{23352535288}{48256443282289} a^{12} + \frac{17932092511358}{48256443282289} a^{11} - \frac{12722914876773}{48256443282289} a^{10} + \frac{8090747044228}{48256443282289} a^{9} + \frac{18747152437503}{48256443282289} a^{8} - \frac{327400370115}{1556659460719} a^{7} + \frac{20193905590181}{48256443282289} a^{6} + \frac{16534279153356}{48256443282289} a^{5} - \frac{13227227642975}{48256443282289} a^{4} - \frac{23212761215707}{48256443282289} a^{3} + \frac{682912560872}{48256443282289} a^{2} - \frac{10979054517740}{48256443282289} a - \frac{3293353698507}{48256443282289}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 73205.1735676 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times D_7$ (as 14T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 98
The 35 conjugacy class representatives for $C_7 \wr C_2$
Character table for $C_7 \wr C_2$ is not computed

Intermediate fields

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

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
7Data not computed