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Properties

Label 15.3.30448615340...1921.1
Degree $15$
Signature $[3, 6]$
Discriminant $7^{10}\cdot 47^{6}$
Root discriminant $17.07$
Ramified primes $7, 47$
Class number $1$
Class group Trivial
Galois group $D_5\times C_3$ (as 15T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 6, -22, 47, -39, -37, 37, 2, 37, 12, -2, 0, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^11 + 12*x^10 + 37*x^9 + 2*x^8 + 37*x^7 - 37*x^6 - 39*x^5 + 47*x^4 - 22*x^3 + 6*x^2 - 1)
 
gp: K = bnfinit(x^15 - 2*x^14 + 2*x^13 - 2*x^11 + 12*x^10 + 37*x^9 + 2*x^8 + 37*x^7 - 37*x^6 - 39*x^5 + 47*x^4 - 22*x^3 + 6*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} + 2 x^{13} - 2 x^{11} + 12 x^{10} + 37 x^{9} + 2 x^{8} + 37 x^{7} - 37 x^{6} - 39 x^{5} + 47 x^{4} - 22 x^{3} + 6 x^{2} - 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3044861534083891921=7^{10}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.07$
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:  $7, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{83} a^{13} + \frac{38}{83} a^{12} - \frac{40}{83} a^{11} - \frac{34}{83} a^{10} + \frac{30}{83} a^{9} + \frac{38}{83} a^{8} + \frac{15}{83} a^{7} + \frac{10}{83} a^{6} - \frac{2}{83} a^{5} + \frac{33}{83} a^{4} + \frac{6}{83} a^{3} + \frac{35}{83} a^{2} - \frac{26}{83} a - \frac{11}{83}$, $\frac{1}{35841834307} a^{14} - \frac{120538060}{35841834307} a^{13} + \frac{4243453050}{35841834307} a^{12} + \frac{8911653662}{35841834307} a^{11} + \frac{3933592569}{35841834307} a^{10} + \frac{2531369626}{35841834307} a^{9} - \frac{4695103826}{35841834307} a^{8} + \frac{4504985066}{35841834307} a^{7} - \frac{10404426607}{35841834307} a^{6} + \frac{11482405163}{35841834307} a^{5} - \frac{15433288080}{35841834307} a^{4} - \frac{8951135085}{35841834307} a^{3} + \frac{3046289095}{35841834307} a^{2} + \frac{14480179762}{35841834307} a - \frac{3980446716}{35841834307}$

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:  $8$
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:  \( 1650.39003052 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_5$ (as 15T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 12 conjugacy class representatives for $D_5\times C_3$
Character table for $D_5\times C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.1.2209.1

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 $15$ $15$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ R $15$ $15$

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
7Data not computed
$47$47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$