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Properties

Label 14.0.15480223191...1959.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 17^{7}\cdot 29^{7}$
Root discriminant $38.46$
Ramified primes $3, 17, 29$
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![13311, -53244, 88776, -84987, 58806, -34815, 16555, -4512, -451, 846, -225, 3, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 23*x^12 + 3*x^11 - 225*x^10 + 846*x^9 - 451*x^8 - 4512*x^7 + 16555*x^6 - 34815*x^5 + 58806*x^4 - 84987*x^3 + 88776*x^2 - 53244*x + 13311)
 
gp: K = bnfinit(x^14 - 6*x^13 + 23*x^12 + 3*x^11 - 225*x^10 + 846*x^9 - 451*x^8 - 4512*x^7 + 16555*x^6 - 34815*x^5 + 58806*x^4 - 84987*x^3 + 88776*x^2 - 53244*x + 13311, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 23 x^{12} + 3 x^{11} - 225 x^{10} + 846 x^{9} - 451 x^{8} - 4512 x^{7} + 16555 x^{6} - 34815 x^{5} + 58806 x^{4} - 84987 x^{3} + 88776 x^{2} - 53244 x + 13311 \)

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:  \(-15480223191305595931959=-\,3^{7}\cdot 17^{7}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.46$
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:  $3, 17, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{63} a^{11} + \frac{1}{21} a^{10} - \frac{1}{7} a^{8} + \frac{1}{21} a^{7} - \frac{16}{63} a^{5} - \frac{1}{3} a^{4} + \frac{10}{21} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{3213} a^{12} + \frac{8}{1071} a^{11} + \frac{23}{459} a^{10} + \frac{32}{1071} a^{9} + \frac{64}{1071} a^{8} - \frac{352}{3213} a^{6} - \frac{11}{153} a^{5} - \frac{46}{189} a^{4} - \frac{281}{1071} a^{3} - \frac{31}{1071} a^{2} + \frac{5}{21} a - \frac{1}{3}$, $\frac{1}{12386089650716160687} a^{13} - \frac{116689643013827}{4128696550238720229} a^{12} + \frac{16128272608984433}{12386089650716160687} a^{11} + \frac{37301547823254328}{4128696550238720229} a^{10} - \frac{3288303545158850}{589813792891245747} a^{9} - \frac{11838531757344314}{125112016673900613} a^{8} - \frac{81737370821347295}{1126008150065105517} a^{7} - \frac{348490468095714568}{4128696550238720229} a^{6} + \frac{329523579773986276}{1769441378673737241} a^{5} + \frac{294023704405992578}{589813792891245747} a^{4} - \frac{806805554851237765}{4128696550238720229} a^{3} + \frac{29386358759977202}{458744061137635581} a^{2} - \frac{27274057231897750}{80954834318406279} a + \frac{1786477861150469}{26984944772802093}$

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:  \( 718479.336911 \) (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

\(\Q(\sqrt{-1479}) \), 7.1.3235225239.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.3235225239.1

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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$