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Properties

Label 14.0.13805144387...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 97^{7}$
Root discriminant $38.14$
Ramified primes $3, 5, 97$
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![13995, 6420, 6184, 12942, 12843, -4242, 1134, 816, 358, 261, -63, -24, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 24*x^11 - 63*x^10 + 261*x^9 + 358*x^8 + 816*x^7 + 1134*x^6 - 4242*x^5 + 12843*x^4 + 12942*x^3 + 6184*x^2 + 6420*x + 13995)
 
gp: K = bnfinit(x^14 - 24*x^11 - 63*x^10 + 261*x^9 + 358*x^8 + 816*x^7 + 1134*x^6 - 4242*x^5 + 12843*x^4 + 12942*x^3 + 6184*x^2 + 6420*x + 13995, 1)
 

Normalized defining polynomial

\( x^{14} - 24 x^{11} - 63 x^{10} + 261 x^{9} + 358 x^{8} + 816 x^{7} + 1134 x^{6} - 4242 x^{5} + 12843 x^{4} + 12942 x^{3} + 6184 x^{2} + 6420 x + 13995 \)

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:  \(-13805144387002588359375=-\,3^{7}\cdot 5^{7}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.14$
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:  $3, 5, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{45} a^{8} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{2}{15} a^{5} - \frac{4}{15} a^{4} - \frac{4}{15} a^{3} - \frac{19}{45} a^{2} - \frac{1}{3} a$, $\frac{1}{45} a^{9} - \frac{2}{15} a^{7} - \frac{1}{15} a^{6} - \frac{7}{15} a^{4} + \frac{2}{45} a^{3} - \frac{1}{15} a^{2} - \frac{1}{3} a$, $\frac{1}{45} a^{10} + \frac{1}{15} a^{6} + \frac{1}{9} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{675} a^{11} - \frac{1}{675} a^{10} - \frac{4}{675} a^{9} + \frac{7}{675} a^{8} + \frac{1}{225} a^{7} - \frac{2}{15} a^{6} + \frac{62}{675} a^{5} + \frac{29}{135} a^{4} - \frac{176}{675} a^{3} - \frac{277}{675} a^{2} + \frac{1}{45} a + \frac{7}{15}$, $\frac{1}{675} a^{12} - \frac{1}{135} a^{10} + \frac{1}{225} a^{9} - \frac{1}{135} a^{8} + \frac{31}{225} a^{7} - \frac{73}{675} a^{6} - \frac{4}{25} a^{5} + \frac{149}{675} a^{4} + \frac{59}{225} a^{3} + \frac{23}{675} a^{2} + \frac{7}{45} a + \frac{7}{15}$, $\frac{1}{23388942016553464125} a^{13} - \frac{13703772435699094}{23388942016553464125} a^{12} - \frac{5411212426693694}{23388942016553464125} a^{11} + \frac{216464961375902072}{23388942016553464125} a^{10} - \frac{700210359030053}{241123113572716125} a^{9} + \frac{27573321617110657}{4677788403310692825} a^{8} - \frac{3099684369548213212}{23388942016553464125} a^{7} + \frac{297592157394104884}{23388942016553464125} a^{6} - \frac{467267925322005847}{23388942016553464125} a^{5} + \frac{3607031079622480111}{23388942016553464125} a^{4} + \frac{1874642171423112239}{23388942016553464125} a^{3} - \frac{6285809513152673069}{23388942016553464125} a^{2} - \frac{502524026176602121}{1559262801103564275} a + \frac{223817356651741784}{519754267034521425}$

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:  \( 1591509.23851 \) (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

\(\Q(\sqrt{-1455}) \), 7.1.3080271375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.3080271375.1

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 R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$$[\ ]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$