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Properties

Label 14.0.66560847311...2271.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 19^{7}\cdot 23^{7}$
Root discriminant $36.21$
Ramified primes $3, 19, 23$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3969, 2079, 12078, -10242, 4557, -2313, 2020, -10, -689, 412, -52, -23, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 16*x^12 - 23*x^11 - 52*x^10 + 412*x^9 - 689*x^8 - 10*x^7 + 2020*x^6 - 2313*x^5 + 4557*x^4 - 10242*x^3 + 12078*x^2 + 2079*x + 3969)
 
gp: K = bnfinit(x^14 - 4*x^13 + 16*x^12 - 23*x^11 - 52*x^10 + 412*x^9 - 689*x^8 - 10*x^7 + 2020*x^6 - 2313*x^5 + 4557*x^4 - 10242*x^3 + 12078*x^2 + 2079*x + 3969, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 16 x^{12} - 23 x^{11} - 52 x^{10} + 412 x^{9} - 689 x^{8} - 10 x^{7} + 2020 x^{6} - 2313 x^{5} + 4557 x^{4} - 10242 x^{3} + 12078 x^{2} + 2079 x + 3969 \)

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:  \(-6656084731100035682271=-\,3^{7}\cdot 19^{7}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.21$
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:  $3, 19, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{189} a^{11} - \frac{5}{189} a^{10} - \frac{10}{189} a^{9} - \frac{5}{189} a^{8} + \frac{11}{189} a^{7} + \frac{31}{189} a^{6} - \frac{32}{189} a^{5} + \frac{16}{63} a^{4} + \frac{29}{63} a^{3} + \frac{2}{21} a^{2} - \frac{1}{7} a$, $\frac{1}{3591} a^{12} + \frac{1}{3591} a^{11} + \frac{86}{3591} a^{10} + \frac{187}{3591} a^{9} - \frac{82}{3591} a^{8} - \frac{155}{3591} a^{7} + \frac{31}{513} a^{6} - \frac{107}{399} a^{5} + \frac{293}{1197} a^{4} - \frac{80}{399} a^{3} + \frac{191}{399} a^{2} + \frac{36}{133} a - \frac{5}{19}$, $\frac{1}{224866681373335113} a^{13} + \frac{6622018603003}{224866681373335113} a^{12} + \frac{169454274056437}{224866681373335113} a^{11} + \frac{276787871769683}{8328395606419819} a^{10} - \frac{247642141252652}{24985186819259457} a^{9} + \frac{66097378974616}{3569312402751351} a^{8} + \frac{12474019142738606}{224866681373335113} a^{7} - \frac{767340418036798}{32123811624762159} a^{6} + \frac{8217161309597}{224866681373335113} a^{5} + \frac{28195592231636954}{74955560457778371} a^{4} + \frac{12191599465470287}{74955560457778371} a^{3} - \frac{1286677481117848}{8328395606419819} a^{2} + \frac{9695142707471708}{24985186819259457} a + \frac{91122411966354}{1189770800917117}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 468416.967019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-1311}) \), 7.1.2253243231.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.2253243231.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$