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Properties

Label 15.1.72459688058...3103.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1327^{7}$
Root discriminant $28.66$
Ramified prime $1327$
Class number $1$
Class group Trivial
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 83, -21, -18, 100, 13, -20, 52, 3, 24, 1, 38, -24, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1)
 
gp: K = bnfinit(x^15 - 6*x^14 + 16*x^13 - 24*x^12 + 38*x^11 + x^10 + 24*x^9 + 3*x^8 + 52*x^7 - 20*x^6 + 13*x^5 + 100*x^4 - 18*x^3 - 21*x^2 + 83*x + 1, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 16 x^{13} - 24 x^{12} + 38 x^{11} + x^{10} + 24 x^{9} + 3 x^{8} + 52 x^{7} - 20 x^{6} + 13 x^{5} + 100 x^{4} - 18 x^{3} - 21 x^{2} + 83 x + 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7245968805874891233103=-\,1327^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.66$
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:  $1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{8} - \frac{2}{15} a^{7} - \frac{7}{15} a^{6} + \frac{4}{15} a^{5} + \frac{2}{15} a^{4} + \frac{7}{15} a^{3} - \frac{7}{15} a^{2} + \frac{4}{15} a + \frac{4}{15}$, $\frac{1}{15} a^{9} - \frac{1}{15} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{15} a^{4} + \frac{2}{15} a^{3} - \frac{1}{3} a^{2} + \frac{7}{15} a - \frac{2}{15}$, $\frac{1}{45} a^{10} + \frac{1}{45} a^{9} + \frac{2}{45} a^{7} - \frac{2}{45} a^{6} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{45} a^{3} + \frac{19}{45} a + \frac{7}{45}$, $\frac{1}{45} a^{11} - \frac{1}{45} a^{9} - \frac{1}{45} a^{8} + \frac{2}{45} a^{7} + \frac{13}{45} a^{6} + \frac{8}{45} a^{5} - \frac{17}{45} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{7}{15} a - \frac{19}{45}$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{8} + \frac{2}{15} a^{7} - \frac{1}{15} a^{6} - \frac{1}{5} a^{5} - \frac{1}{45} a^{4} + \frac{1}{15} a^{3} + \frac{4}{15} a^{2} + \frac{2}{5} a + \frac{2}{9}$, $\frac{1}{2025} a^{13} - \frac{19}{2025} a^{12} - \frac{4}{405} a^{11} - \frac{4}{675} a^{10} + \frac{49}{2025} a^{9} + \frac{1}{135} a^{8} - \frac{178}{2025} a^{7} + \frac{187}{2025} a^{6} + \frac{37}{81} a^{5} + \frac{464}{2025} a^{4} - \frac{719}{2025} a^{3} - \frac{37}{405} a^{2} + \frac{979}{2025} a + \frac{187}{2025}$, $\frac{1}{115425} a^{14} + \frac{1}{6075} a^{13} + \frac{833}{115425} a^{12} - \frac{1087}{115425} a^{11} - \frac{632}{115425} a^{10} + \frac{3182}{115425} a^{9} + \frac{1427}{115425} a^{8} + \frac{7463}{115425} a^{7} - \frac{15368}{38475} a^{6} - \frac{47276}{115425} a^{5} + \frac{16823}{115425} a^{4} - \frac{3529}{38475} a^{3} + \frac{5078}{38475} a^{2} - \frac{1022}{38475} a + \frac{42791}{115425}$

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:  $7$
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:  \( 225706.053096 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.1327.1, 5.1.1760929.1

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

Sibling fields

Galois closure: 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 $15$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1327Data not computed