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Properties

Label 15.15.1084436136...8009.1
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![-20691, -173151, -509652, -586080, -64647, 396594, 259443, -28503, -74160, -15385, 5091, 1827, -104, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 72*x^13 - 104*x^12 + 1827*x^11 + 5091*x^10 - 15385*x^9 - 74160*x^8 - 28503*x^7 + 259443*x^6 + 396594*x^5 - 64647*x^4 - 586080*x^3 - 509652*x^2 - 173151*x - 20691)
 
gp: K = bnfinit(x^15 - 72*x^13 - 104*x^12 + 1827*x^11 + 5091*x^10 - 15385*x^9 - 74160*x^8 - 28503*x^7 + 259443*x^6 + 396594*x^5 - 64647*x^4 - 586080*x^3 - 509652*x^2 - 173151*x - 20691, 1)
 

Normalized defining polynomial

\( x^{15} - 72 x^{13} - 104 x^{12} + 1827 x^{11} + 5091 x^{10} - 15385 x^{9} - 74160 x^{8} - 28503 x^{7} + 259443 x^{6} + 396594 x^{5} - 64647 x^{4} - 586080 x^{3} - 509652 x^{2} - 173151 x - 20691 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{561} a^{12} + \frac{1}{17} a^{11} + \frac{86}{187} a^{10} - \frac{236}{561} a^{9} + \frac{15}{187} a^{8} - \frac{52}{187} a^{7} - \frac{7}{561} a^{6} + \frac{8}{187} a^{5} + \frac{69}{187} a^{4} - \frac{89}{187} a^{3} - \frac{3}{17} a^{2} - \frac{2}{17} a - \frac{8}{17}$, $\frac{1}{561} a^{13} - \frac{83}{561} a^{11} + \frac{226}{561} a^{10} - \frac{7}{187} a^{9} + \frac{229}{561} a^{8} + \frac{92}{561} a^{7} + \frac{5}{11} a^{6} - \frac{211}{561} a^{5} + \frac{65}{187} a^{4} - \frac{8}{17} a^{3} - \frac{5}{17} a^{2} + \frac{7}{17} a - \frac{8}{17}$, $\frac{1}{17602909362074859141} a^{14} - \frac{2179350105313775}{17602909362074859141} a^{13} + \frac{543241229867999}{1035465256592638773} a^{12} - \frac{2232148685281076513}{17602909362074859141} a^{11} - \frac{6741843861089124248}{17602909362074859141} a^{10} + \frac{8745415879267470085}{17602909362074859141} a^{9} - \frac{3061091455081215631}{17602909362074859141} a^{8} + \frac{4845500242804298864}{17602909362074859141} a^{7} + \frac{8395893315810445895}{17602909362074859141} a^{6} - \frac{2738911532471520396}{5867636454024953047} a^{5} - \frac{538183247878163455}{5867636454024953047} a^{4} - \frac{2708918755606789046}{5867636454024953047} a^{3} - \frac{125656096581405118}{533421495820450277} a^{2} + \frac{38208214104996952}{533421495820450277} a + \frac{6584763150476382}{28074815569497383}$

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}$