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Properties

Label 14.10.2148755578...8912.1
Degree $14$
Signature $[10, 2]$
Discriminant $2^{31}\cdot 23^{2}\cdot 73^{2}\cdot 59576897^{2}$
Root discriminant $172.99$
Ramified primes $2, 23, 73, 59576897$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T58

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4574, 14280, -8061, -16792, 28590, -14484, -4934, 8476, -1404, -1576, 432, 132, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 36*x^12 + 132*x^11 + 432*x^10 - 1576*x^9 - 1404*x^8 + 8476*x^7 - 4934*x^6 - 14484*x^5 + 28590*x^4 - 16792*x^3 - 8061*x^2 + 14280*x - 4574)
 
gp: K = bnfinit(x^14 - 4*x^13 - 36*x^12 + 132*x^11 + 432*x^10 - 1576*x^9 - 1404*x^8 + 8476*x^7 - 4934*x^6 - 14484*x^5 + 28590*x^4 - 16792*x^3 - 8061*x^2 + 14280*x - 4574, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 36 x^{12} + 132 x^{11} + 432 x^{10} - 1576 x^{9} - 1404 x^{8} + 8476 x^{7} - 4934 x^{6} - 14484 x^{5} + 28590 x^{4} - 16792 x^{3} - 8061 x^{2} + 14280 x - 4574 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21487555788040560033827139878912=2^{31}\cdot 23^{2}\cdot 73^{2}\cdot 59576897^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.99$
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:  $2, 23, 73, 59576897$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{2667} a^{12} + \frac{61}{889} a^{11} - \frac{86}{2667} a^{10} + \frac{100}{2667} a^{9} - \frac{4}{381} a^{8} + \frac{41}{2667} a^{7} + \frac{1159}{2667} a^{6} - \frac{1088}{2667} a^{5} - \frac{21}{127} a^{4} + \frac{106}{2667} a^{3} + \frac{790}{2667} a^{2} - \frac{17}{2667} a - \frac{40}{2667}$, $\frac{1}{397054959} a^{13} + \frac{14908}{397054959} a^{12} + \frac{4054378}{397054959} a^{11} - \frac{2136076}{56722137} a^{10} - \frac{882300}{132351653} a^{9} + \frac{20687521}{397054959} a^{8} + \frac{7481014}{132351653} a^{7} - \frac{146427247}{397054959} a^{6} - \frac{149901212}{397054959} a^{5} + \frac{8135677}{56722137} a^{4} + \frac{96552941}{397054959} a^{3} + \frac{140621807}{397054959} a^{2} + \frac{40009048}{132351653} a - \frac{55595875}{397054959}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 647005758978 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T58:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 12700800
The 54 conjugacy class representatives for [A(7)^2]2=A(7)wr2 are not computed
Character table for [A(7)^2]2=A(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 30 siblings: data not computed
Degree 42 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
73Data not computed
59576897Data not computed