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Properties

Label 15.1.13473473081...2751.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,751^{7}$
Root discriminant $21.98$
Ramified prime $751$
Class number $2$
Class group $[2]$
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, 25, 12, 53, -9, 21, -4, 2, -29, 17, -12, 7, -6, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 9*x^13 - 6*x^12 + 7*x^11 - 12*x^10 + 17*x^9 - 29*x^8 + 2*x^7 - 4*x^6 + 21*x^5 - 9*x^4 + 53*x^3 + 12*x^2 + 25*x - 1)
 
gp: K = bnfinit(x^15 - 5*x^14 + 9*x^13 - 6*x^12 + 7*x^11 - 12*x^10 + 17*x^9 - 29*x^8 + 2*x^7 - 4*x^6 + 21*x^5 - 9*x^4 + 53*x^3 + 12*x^2 + 25*x - 1, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 9 x^{13} - 6 x^{12} + 7 x^{11} - 12 x^{10} + 17 x^{9} - 29 x^{8} + 2 x^{7} - 4 x^{6} + 21 x^{5} - 9 x^{4} + 53 x^{3} + 12 x^{2} + 25 x - 1 \)

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:  \(-134734730815558692751=-\,751^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.98$
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:  $751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{783} a^{13} + \frac{4}{87} a^{12} + \frac{68}{783} a^{11} + \frac{55}{783} a^{10} + \frac{127}{783} a^{9} + \frac{79}{783} a^{8} - \frac{2}{261} a^{7} + \frac{4}{261} a^{6} + \frac{383}{783} a^{5} + \frac{1}{3} a^{4} - \frac{332}{783} a^{3} - \frac{310}{783} a^{2} + \frac{254}{783} a + \frac{254}{783}$, $\frac{1}{1237923} a^{14} - \frac{670}{1237923} a^{13} - \frac{28741}{1237923} a^{12} - \frac{25246}{1237923} a^{11} - \frac{65362}{412641} a^{10} - \frac{29165}{412641} a^{9} + \frac{11297}{1237923} a^{8} - \frac{46334}{137547} a^{7} + \frac{384194}{1237923} a^{6} + \frac{21139}{1237923} a^{5} - \frac{122480}{1237923} a^{4} + \frac{571555}{1237923} a^{3} - \frac{22547}{137547} a^{2} + \frac{2804}{24273} a + \frac{241408}{1237923}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 10575.7740384 \)
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

3.1.751.1, 5.1.564001.1

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 $15$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ $15$

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
751Data not computed