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Properties

Label 15.15.1084436136...8009.2
Degree $15$
Signature $[15, 0]$
Discriminant $3^{24}\cdot 11^{10}\cdot 23^{6}$
Root discriminant $100.54$
Ramified primes $3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\GL(2,4)$ (as 15T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, -9438, 22869, 60236, -53361, -112068, 32521, 74394, -6696, -19165, 525, 1827, -13, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 72*x^13 - 13*x^12 + 1827*x^11 + 525*x^10 - 19165*x^9 - 6696*x^8 + 74394*x^7 + 32521*x^6 - 112068*x^5 - 53361*x^4 + 60236*x^3 + 22869*x^2 - 9438*x - 121)
 
gp: K = bnfinit(x^15 - 72*x^13 - 13*x^12 + 1827*x^11 + 525*x^10 - 19165*x^9 - 6696*x^8 + 74394*x^7 + 32521*x^6 - 112068*x^5 - 53361*x^4 + 60236*x^3 + 22869*x^2 - 9438*x - 121, 1)
 

Normalized defining polynomial

\( x^{15} - 72 x^{13} - 13 x^{12} + 1827 x^{11} + 525 x^{10} - 19165 x^{9} - 6696 x^{8} + 74394 x^{7} + 32521 x^{6} - 112068 x^{5} - 53361 x^{4} + 60236 x^{3} + 22869 x^{2} - 9438 x - 121 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1084436136203703749597157948009=3^{24}\cdot 11^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.54$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{33} a^{12} - \frac{2}{11} a^{10} - \frac{2}{33} a^{9} + \frac{4}{11} a^{8} - \frac{1}{11} a^{7} - \frac{14}{33} a^{6} + \frac{1}{11} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a^{3} + \frac{1}{3}$, $\frac{1}{33} a^{13} + \frac{5}{33} a^{11} - \frac{13}{33} a^{10} - \frac{10}{33} a^{9} - \frac{1}{11} a^{8} - \frac{14}{33} a^{7} + \frac{1}{11} a^{6} - \frac{10}{33} a^{5} + \frac{16}{33} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{2444083161593464364926953} a^{14} + \frac{9845187460303558671248}{2444083161593464364926953} a^{13} + \frac{2393467504826593698370}{349154737370494909275279} a^{12} + \frac{156945247112478724308382}{2444083161593464364926953} a^{11} - \frac{1200937534615733988592348}{2444083161593464364926953} a^{10} - \frac{235231821779084117922842}{2444083161593464364926953} a^{9} - \frac{5572552029177151757239}{31741339760954082661389} a^{8} + \frac{633580577185668708045614}{2444083161593464364926953} a^{7} + \frac{7395299566465084805795}{31741339760954082661389} a^{6} + \frac{12765138240396314677277}{814694387197821454975651} a^{5} + \frac{365352037529980457508239}{814694387197821454975651} a^{4} + \frac{184081209628818473995652}{814694387197821454975651} a^{3} + \frac{58776589478421810600967}{222189378326678578629723} a^{2} + \frac{46834320066385082367515}{222189378326678578629723} a - \frac{64555723591528511238107}{222189378326678578629723}$

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:  $14$
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:  \( 43649743656.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_5$ (as 15T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 180
The 15 conjugacy class representatives for $\GL(2,4)$
Character table for $\GL(2,4)$

Intermediate fields

5.5.5184729.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 18 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: data not computed
Arithmetically equvalently 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 $15$ R $15$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ R $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$11$11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$23$23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$