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Properties

Label 14.6.55179078052...6381.1
Degree $14$
Signature $[6, 4]$
Discriminant $3^{19}\cdot 7^{15}$
Root discriminant $35.73$
Ramified primes $3, 7$
Class number $1$
Class group Trivial
Galois group $SO(3,7)$ (as 14T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-68, -322, 49, 140, -203, 308, 21, -309, 147, 35, -77, 35, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 7*x^12 + 35*x^11 - 77*x^10 + 35*x^9 + 147*x^8 - 309*x^7 + 21*x^6 + 308*x^5 - 203*x^4 + 140*x^3 + 49*x^2 - 322*x - 68)
 
gp: K = bnfinit(x^14 - 7*x^13 + 7*x^12 + 35*x^11 - 77*x^10 + 35*x^9 + 147*x^8 - 309*x^7 + 21*x^6 + 308*x^5 - 203*x^4 + 140*x^3 + 49*x^2 - 322*x - 68, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 7 x^{12} + 35 x^{11} - 77 x^{10} + 35 x^{9} + 147 x^{8} - 309 x^{7} + 21 x^{6} + 308 x^{5} - 203 x^{4} + 140 x^{3} + 49 x^{2} - 322 x - 68 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5517907805219086266381=3^{19}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.73$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{135} a^{11} + \frac{1}{27} a^{10} + \frac{19}{135} a^{9} - \frac{7}{45} a^{8} + \frac{8}{45} a^{7} - \frac{1}{3} a^{6} + \frac{2}{5} a^{5} - \frac{14}{45} a^{4} - \frac{1}{9} a^{3} + \frac{38}{135} a^{2} - \frac{14}{135} a + \frac{23}{135}$, $\frac{1}{14985} a^{12} - \frac{38}{14985} a^{11} + \frac{29}{14985} a^{10} + \frac{1772}{14985} a^{9} - \frac{32}{1665} a^{8} - \frac{2279}{4995} a^{7} - \frac{53}{555} a^{6} - \frac{473}{4995} a^{5} - \frac{2}{555} a^{4} + \frac{6983}{14985} a^{3} + \frac{1592}{14985} a^{2} + \frac{62}{2997} a - \frac{7424}{14985}$, $\frac{1}{14415570} a^{13} + \frac{107}{14415570} a^{12} + \frac{5311}{1601730} a^{11} + \frac{252397}{14415570} a^{10} + \frac{974267}{14415570} a^{9} + \frac{63721}{4805190} a^{8} + \frac{323413}{4805190} a^{7} + \frac{1880077}{4805190} a^{6} - \frac{1264073}{4805190} a^{5} - \frac{1631291}{7207785} a^{4} + \frac{3831307}{14415570} a^{3} + \frac{7654}{36963} a^{2} + \frac{1014881}{14415570} a + \frac{275635}{1441557}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 2315086.73291 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SO(3,7)$ (as 14T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $SO(3,7)$
Character table for $SO(3,7)$

Intermediate fields

\(\Q(\sqrt{21}) \)

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

Sibling fields

Degree 8 sibling: data not computed
Degree 16 sibling: data not computed
Degree 21 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.9.4$x^{6} + 6 x^{4} + 6$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
3.6.9.4$x^{6} + 6 x^{4} + 6$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
7Data not computed