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Properties

Label 14.6.75728092728...5161.1
Degree $14$
Signature $[6, 4]$
Discriminant $7^{10}\cdot 173^{6}$
Root discriminant $36.54$
Ramified primes $7, 173$
Class number $1$
Class group Trivial
Galois group 14T41

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -37, -45, 115, -65, -333, 359, 74, -263, 60, 2, -3, 12, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 12*x^12 - 3*x^11 + 2*x^10 + 60*x^9 - 263*x^8 + 74*x^7 + 359*x^6 - 333*x^5 - 65*x^4 + 115*x^3 - 45*x^2 - 37*x - 4)
 
gp: K = bnfinit(x^14 - 7*x^13 + 12*x^12 - 3*x^11 + 2*x^10 + 60*x^9 - 263*x^8 + 74*x^7 + 359*x^6 - 333*x^5 - 65*x^4 + 115*x^3 - 45*x^2 - 37*x - 4, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 12 x^{12} - 3 x^{11} + 2 x^{10} + 60 x^{9} - 263 x^{8} + 74 x^{7} + 359 x^{6} - 333 x^{5} - 65 x^{4} + 115 x^{3} - 45 x^{2} - 37 x - 4 \)

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:  \(7572809272861419965161=7^{10}\cdot 173^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.54$
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:  $7, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{675695665858249} a^{13} - \frac{26725375229767}{675695665858249} a^{12} - \frac{207343585437324}{675695665858249} a^{11} + \frac{80150919900516}{675695665858249} a^{10} + \frac{209907067749927}{675695665858249} a^{9} + \frac{248707269361971}{675695665858249} a^{8} - \frac{111993836645147}{675695665858249} a^{7} - \frac{190412625695708}{675695665858249} a^{6} + \frac{211026312934569}{675695665858249} a^{5} + \frac{207993445046022}{675695665858249} a^{4} + \frac{31132791003890}{675695665858249} a^{3} - \frac{186749849492305}{675695665858249} a^{2} + \frac{202915934603468}{675695665858249} a - \frac{105778681057093}{675695665858249}$

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:  \( 1701246.19753 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T41:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 20 conjugacy class representatives for [2^6]F_42(7)
Character table for [2^6]F_42(7)

Intermediate fields

7.7.12431698517.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$173$$\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
173.12.6.1$x^{12} + 196753246 x^{6} - 154963892093 x^{2} + 9677959952884129$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$