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Properties

Label 14.0.57616882545...8688.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 13^{10}$
Root discriminant $42.24$
Ramified primes $2, 3, 13$
Class number $7$ (GRH)
Class group $[7]$ (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![16, -92, 625, 344, 1118, -16, 789, -18, 336, -46, 83, -10, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 13*x^12 - 10*x^11 + 83*x^10 - 46*x^9 + 336*x^8 - 18*x^7 + 789*x^6 - 16*x^5 + 1118*x^4 + 344*x^3 + 625*x^2 - 92*x + 16)
 
gp: K = bnfinit(x^14 - 2*x^13 + 13*x^12 - 10*x^11 + 83*x^10 - 46*x^9 + 336*x^8 - 18*x^7 + 789*x^6 - 16*x^5 + 1118*x^4 + 344*x^3 + 625*x^2 - 92*x + 16, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 13 x^{12} - 10 x^{11} + 83 x^{10} - 46 x^{9} + 336 x^{8} - 18 x^{7} + 789 x^{6} - 16 x^{5} + 1118 x^{4} + 344 x^{3} + 625 x^{2} - 92 x + 16 \)

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:  \(-57616882545504610418688=-\,2^{18}\cdot 3^{13}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.24$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{4594020537132} a^{13} + \frac{12595890371}{255223363174} a^{12} - \frac{73935142861}{1531340179044} a^{11} - \frac{11956091909}{2297010268566} a^{10} + \frac{167718196733}{1531340179044} a^{9} + \frac{80128869295}{765670089522} a^{8} + \frac{14369966216}{127611681587} a^{7} + \frac{374091385651}{765670089522} a^{6} + \frac{181651257329}{510446726348} a^{5} + \frac{560603561918}{1148505134283} a^{4} - \frac{10620712399}{255223363174} a^{3} - \frac{22392856306}{382835044761} a^{2} - \frac{460790683319}{4594020537132} a + \frac{173740453301}{382835044761}$

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:  \( -\frac{15859366663}{1531340179044} a^{13} + \frac{15322876633}{765670089522} a^{12} - \frac{67288589769}{510446726348} a^{11} + \frac{69856976009}{765670089522} a^{10} - \frac{426206089919}{510446726348} a^{9} + \frac{307507718165}{765670089522} a^{8} - \frac{1279600123976}{382835044761} a^{7} - \frac{109419691495}{765670089522} a^{6} - \frac{11880592619627}{1531340179044} a^{5} - \frac{199822414138}{382835044761} a^{4} - \frac{8244783546413}{765670089522} a^{3} - \frac{1912895678621}{382835044761} a^{2} - \frac{2932747422597}{510446726348} a + \frac{322952553577}{382835044761} \) (order $6$)
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:  \( 344206.360414 \) (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

\(\Q(\sqrt{-3}) \), 7.7.138584369664.1

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$