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Properties

Label 14.0.84856922099...4287.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,19^{7}\cdot 37^{7}$
Root discriminant $26.51$
Ramified primes $19, 37$
Class number $2$
Class group $[2]$
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![25, -175, 2028, -755, 4247, -702, 2542, -148, 637, -54, 137, -32, 24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 24*x^12 - 32*x^11 + 137*x^10 - 54*x^9 + 637*x^8 - 148*x^7 + 2542*x^6 - 702*x^5 + 4247*x^4 - 755*x^3 + 2028*x^2 - 175*x + 25)
 
gp: K = bnfinit(x^14 - 4*x^13 + 24*x^12 - 32*x^11 + 137*x^10 - 54*x^9 + 637*x^8 - 148*x^7 + 2542*x^6 - 702*x^5 + 4247*x^4 - 755*x^3 + 2028*x^2 - 175*x + 25, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 24 x^{12} - 32 x^{11} + 137 x^{10} - 54 x^{9} + 637 x^{8} - 148 x^{7} + 2542 x^{6} - 702 x^{5} + 4247 x^{4} - 755 x^{3} + 2028 x^{2} - 175 x + 25 \)

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:  \(-84856922099409044287=-\,19^{7}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.51$
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:  $19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{15} a^{9} - \frac{2}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{6} + \frac{1}{15} a^{5} + \frac{1}{15} a^{4} + \frac{7}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{5} a$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{8} + \frac{2}{15} a^{7} - \frac{4}{15} a^{6} - \frac{7}{15} a^{5} - \frac{1}{15} a^{4} - \frac{7}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{15} a$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{7}{15} a^{6} + \frac{2}{15} a^{5} - \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{15} a^{2} - \frac{7}{15} a - \frac{1}{3}$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{11} - \frac{1}{45} a^{10} - \frac{1}{45} a^{9} - \frac{1}{9} a^{7} + \frac{13}{45} a^{6} + \frac{19}{45} a^{5} + \frac{2}{5} a^{4} - \frac{13}{45} a^{3} - \frac{13}{45} a^{2} - \frac{7}{45} a + \frac{2}{9}$, $\frac{1}{37491923882880315} a^{13} - \frac{96119107508311}{37491923882880315} a^{12} - \frac{319647036920749}{37491923882880315} a^{11} - \frac{380082925088821}{37491923882880315} a^{10} - \frac{3940821632986}{833153864064007} a^{9} + \frac{5542878837332602}{37491923882880315} a^{8} - \frac{6059860418961749}{37491923882880315} a^{7} + \frac{286609882195115}{7498384776576063} a^{6} - \frac{3311787531463886}{12497307960960105} a^{5} + \frac{7827273450277214}{37491923882880315} a^{4} + \frac{8165342963841017}{37491923882880315} a^{3} + \frac{8380318379854001}{37491923882880315} a^{2} - \frac{10791560239584314}{37491923882880315} a + \frac{524986791035089}{2499461592192021}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14857.8395435 \)
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

\(\Q(\sqrt{-703}) \), 7.1.347428927.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.347428927.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$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$$[\ ]_{2}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$