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Properties

Label 14.14.1408531439...5408.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{18}\cdot 3^{8}\cdot 7^{15}\cdot 29^{7}$
Root discriminant $197.85$
Ramified primes $2, 3, 7, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![-2795274, 5217534, 7393869, -3152520, -3976098, 737478, 889644, -79686, -96992, 3864, 5208, -70, -126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 126*x^12 - 70*x^11 + 5208*x^10 + 3864*x^9 - 96992*x^8 - 79686*x^7 + 889644*x^6 + 737478*x^5 - 3976098*x^4 - 3152520*x^3 + 7393869*x^2 + 5217534*x - 2795274)
 
gp: K = bnfinit(x^14 - 126*x^12 - 70*x^11 + 5208*x^10 + 3864*x^9 - 96992*x^8 - 79686*x^7 + 889644*x^6 + 737478*x^5 - 3976098*x^4 - 3152520*x^3 + 7393869*x^2 + 5217534*x - 2795274, 1)
 

Normalized defining polynomial

\( x^{14} - 126 x^{12} - 70 x^{11} + 5208 x^{10} + 3864 x^{9} - 96992 x^{8} - 79686 x^{7} + 889644 x^{6} + 737478 x^{5} - 3976098 x^{4} - 3152520 x^{3} + 7393869 x^{2} + 5217534 x - 2795274 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(140853143950073056750358785425408=2^{18}\cdot 3^{8}\cdot 7^{15}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $197.85$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{108} a^{11} - \frac{1}{36} a^{10} - \frac{1}{54} a^{8} + \frac{1}{12} a^{7} + \frac{1}{36} a^{6} - \frac{1}{6} a^{5} - \frac{1}{18} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6}$, $\frac{1}{15552} a^{12} - \frac{31}{7776} a^{11} - \frac{173}{5184} a^{10} - \frac{343}{7776} a^{9} + \frac{2071}{15552} a^{8} - \frac{49}{432} a^{7} - \frac{497}{5184} a^{6} + \frac{47}{648} a^{5} - \frac{431}{5184} a^{4} - \frac{175}{432} a^{3} - \frac{163}{576} a^{2} + \frac{185}{864} a + \frac{143}{864}$, $\frac{1}{24031835431432977494962944} a^{13} + \frac{30843402210540194927}{24031835431432977494962944} a^{12} + \frac{63339158627242985480083}{24031835431432977494962944} a^{11} + \frac{797039380466118609730903}{24031835431432977494962944} a^{10} - \frac{520841319681507325979263}{24031835431432977494962944} a^{9} - \frac{2775113333111314385006873}{24031835431432977494962944} a^{8} + \frac{1088229997016972835181555}{8010611810477659164987648} a^{7} + \frac{780083174295425579529563}{8010611810477659164987648} a^{6} - \frac{947543755590145340708119}{8010611810477659164987648} a^{5} - \frac{1298721163012496481389111}{8010611810477659164987648} a^{4} - \frac{518776872459613535385013}{2670203936825886388329216} a^{3} + \frac{505804793602728978880765}{2670203936825886388329216} a^{2} - \frac{91458494014551115978451}{333775492103235798541152} a + \frac{553139612264702903592515}{1335101968412943194164608}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 758587333915 \) (assuming GRH)
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

\(\Q(\sqrt{203}) \)

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7Data not computed
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$