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Properties

Label 15.7.12039008625...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 1154353^{2}$
Root discriminant $254.32$
Ramified primes $2, 5, 13, 1154353$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T67

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![794065, -4270623, -2556360, 5974767, 1394197, -2720445, -353230, 535969, 64680, -52340, -7866, 3558, 321, -117, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 117*x^13 + 321*x^12 + 3558*x^11 - 7866*x^10 - 52340*x^9 + 64680*x^8 + 535969*x^7 - 353230*x^6 - 2720445*x^5 + 1394197*x^4 + 5974767*x^3 - 2556360*x^2 - 4270623*x + 794065)
 
gp: K = bnfinit(x^15 - 2*x^14 - 117*x^13 + 321*x^12 + 3558*x^11 - 7866*x^10 - 52340*x^9 + 64680*x^8 + 535969*x^7 - 353230*x^6 - 2720445*x^5 + 1394197*x^4 + 5974767*x^3 - 2556360*x^2 - 4270623*x + 794065, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 117 x^{13} + 321 x^{12} + 3558 x^{11} - 7866 x^{10} - 52340 x^{9} + 64680 x^{8} + 535969 x^{7} - 353230 x^{6} - 2720445 x^{5} + 1394197 x^{4} + 5974767 x^{3} - 2556360 x^{2} - 4270623 x + 794065 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1203900862506630378212825497600000000=2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 1154353^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.32$
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:  $2, 5, 13, 1154353$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{10} a^{12} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{3}{10} a$, $\frac{1}{1031847702618042925723799381905965267955670} a^{14} - \frac{1352314097845050020143230980197993780079}{1031847702618042925723799381905965267955670} a^{13} + \frac{38302934195220840671119561798292914171707}{1031847702618042925723799381905965267955670} a^{12} - \frac{49212713596988627861069072209726014519616}{515923851309021462861899690952982633977835} a^{11} + \frac{123040232482182461874535880542822510342479}{515923851309021462861899690952982633977835} a^{10} + \frac{179368739713741359057240829780146313189156}{515923851309021462861899690952982633977835} a^{9} + \frac{3120201037665810394439128801996781171120}{103184770261804292572379938190596526795567} a^{8} + \frac{204537063437468665620092712352340720713197}{515923851309021462861899690952982633977835} a^{7} - \frac{336777444663075300315881338254737455227797}{1031847702618042925723799381905965267955670} a^{6} - \frac{30091494904636743627119616663117022548103}{1031847702618042925723799381905965267955670} a^{5} + \frac{323877741218798293986415382531461773052867}{1031847702618042925723799381905965267955670} a^{4} - \frac{242258399378339158708351924241119713106778}{515923851309021462861899690952982633977835} a^{3} + \frac{512490892978389902714956023474012440845109}{1031847702618042925723799381905965267955670} a^{2} + \frac{7655301514355437001017588470811812723279}{1031847702618042925723799381905965267955670} a + \frac{42752244905582511399666841828618491452259}{206369540523608585144759876381193053591134}$

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:  $10$
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:  \( 9036990762680 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T67:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 32 conjugacy class representatives for [1/2.F(5)^3]3
Character table for [1/2.F(5)^3]3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 30 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$$[\ ]^{3}$
2.12.24.231$x^{12} + 24 x^{11} - 26 x^{10} - 20 x^{9} + 6 x^{8} + 24 x^{7} + 16 x^{6} - 16 x^{5} - 4 x^{4} - 16 x^{3} - 8 x^{2} + 16 x - 24$$4$$3$$24$12T55$[2, 2, 3, 3, 3]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
1154353Data not computed