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Properties

Label 15.1.29155999981...8939.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,1619^{7}$
Root discriminant $31.45$
Ramified prime $1619$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![64, -128, -64, 336, 56, -376, -31, 85, 461, 57, 22, 20, 5, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 5*x^13 + 5*x^12 + 20*x^11 + 22*x^10 + 57*x^9 + 461*x^8 + 85*x^7 - 31*x^6 - 376*x^5 + 56*x^4 + 336*x^3 - 64*x^2 - 128*x + 64)
 
gp: K = bnfinit(x^15 - x^14 + 5*x^13 + 5*x^12 + 20*x^11 + 22*x^10 + 57*x^9 + 461*x^8 + 85*x^7 - 31*x^6 - 376*x^5 + 56*x^4 + 336*x^3 - 64*x^2 - 128*x + 64, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 5 x^{13} + 5 x^{12} + 20 x^{11} + 22 x^{10} + 57 x^{9} + 461 x^{8} + 85 x^{7} - 31 x^{6} - 376 x^{5} + 56 x^{4} + 336 x^{3} - 64 x^{2} - 128 x + 64 \)

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:  \(-29155999981005449978939=-\,1619^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.45$
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:  $1619$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{7}{32} a^{5} + \frac{7}{32} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{416} a^{12} - \frac{3}{208} a^{11} - \frac{1}{32} a^{10} - \frac{1}{208} a^{9} + \frac{3}{208} a^{8} - \frac{1}{26} a^{7} + \frac{49}{416} a^{6} + \frac{9}{52} a^{5} + \frac{1}{32} a^{4} - \frac{23}{52} a^{3} + \frac{1}{4} a^{2} - \frac{7}{52} a + \frac{6}{13}$, $\frac{1}{5824} a^{13} - \frac{1}{2912} a^{12} + \frac{41}{5824} a^{11} - \frac{33}{1456} a^{10} - \frac{53}{2912} a^{9} + \frac{1}{728} a^{8} + \frac{141}{5824} a^{7} - \frac{129}{728} a^{6} - \frac{193}{5824} a^{5} - \frac{313}{2912} a^{4} - \frac{535}{1456} a^{3} + \frac{71}{728} a^{2} - \frac{145}{364} a + \frac{37}{182}$, $\frac{1}{1198625792} a^{14} - \frac{18723}{599312896} a^{13} + \frac{1230419}{1198625792} a^{12} + \frac{5949763}{599312896} a^{11} + \frac{11764899}{599312896} a^{10} + \frac{8055147}{149828224} a^{9} + \frac{5376279}{171232256} a^{8} - \frac{17876723}{149828224} a^{7} - \frac{165448515}{1198625792} a^{6} + \frac{10727007}{74914112} a^{5} - \frac{968427}{11525248} a^{4} + \frac{185265}{74914112} a^{3} + \frac{1461529}{4682132} a^{2} + \frac{3029877}{18728528} a + \frac{6250957}{18728528}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1363950.18144 \) (assuming GRH)
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

3.1.1619.1, 5.1.2621161.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1619Data not computed