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Properties

Label 15.3.22710592261...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 1077707^{2}$
Root discriminant $143.58$
Ramified primes $2, 5, 7, 13, 1077707$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7277179, -55625899, 155961176, -204287273, 145415117, -58450939, 11997606, -529425, -182438, -9030, 11410, -1020, 11, -1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - x^13 + 11*x^12 - 1020*x^11 + 11410*x^10 - 9030*x^9 - 182438*x^8 - 529425*x^7 + 11997606*x^6 - 58450939*x^5 + 145415117*x^4 - 204287273*x^3 + 155961176*x^2 - 55625899*x + 7277179)
 
gp: K = bnfinit(x^15 - 6*x^14 - x^13 + 11*x^12 - 1020*x^11 + 11410*x^10 - 9030*x^9 - 182438*x^8 - 529425*x^7 + 11997606*x^6 - 58450939*x^5 + 145415117*x^4 - 204287273*x^3 + 155961176*x^2 - 55625899*x + 7277179, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - x^{13} + 11 x^{12} - 1020 x^{11} + 11410 x^{10} - 9030 x^{9} - 182438 x^{8} - 529425 x^{7} + 11997606 x^{6} - 58450939 x^{5} + 145415117 x^{4} - 204287273 x^{3} + 155961176 x^{2} - 55625899 x + 7277179 \)

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:  \(227105922614218681297997824000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 1077707^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.58$
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:  $2, 5, 7, 13, 1077707$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{1175298907859861126098969564427591884} a^{14} - \frac{93048121385521624956517225983786267}{1175298907859861126098969564427591884} a^{13} - \frac{142002737157560478611557447428343357}{1175298907859861126098969564427591884} a^{12} - \frac{97148522233422256248578441404620027}{587649453929930563049484782213795942} a^{11} + \frac{108901658731992433806286461422052123}{587649453929930563049484782213795942} a^{10} + \frac{19410243354302955098095395772876985}{293824726964965281524742391106897971} a^{9} - \frac{6354010947527731400803552892748285}{293824726964965281524742391106897971} a^{8} + \frac{71270322751590931568334852759759994}{293824726964965281524742391106897971} a^{7} - \frac{132151050221965473645250113959580721}{1175298907859861126098969564427591884} a^{6} + \frac{447110079800456325624933631768454669}{1175298907859861126098969564427591884} a^{5} + \frac{369729586721677112675974619290028313}{1175298907859861126098969564427591884} a^{4} + \frac{101468452406543256145670068321084155}{293824726964965281524742391106897971} a^{3} - \frac{304229271530054384986445324113999385}{1175298907859861126098969564427591884} a^{2} - \frac{405380357391026850966613169267377279}{1175298907859861126098969564427591884} a - \frac{409226719994789836268606339888414465}{1175298907859861126098969564427591884}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8187698872.63 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(5)wr3 is not computed

Intermediate fields

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

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R $15$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.4$x^{6} + 4 x^{2} + 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
1077707Data not computed