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Properties

Label 15.7.10467392050...6864.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 11^{6}\cdot 79^{5}\cdot 10698269^{2}$
Root discriminant $293.76$
Ramified primes $2, 11, 79, 10698269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12648512, 24462336, 24647840, 10752864, -574336, -2667392, -1075224, -228520, -97106, -38432, 327, 3979, 454, -120, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 120*x^13 + 454*x^12 + 3979*x^11 + 327*x^10 - 38432*x^9 - 97106*x^8 - 228520*x^7 - 1075224*x^6 - 2667392*x^5 - 574336*x^4 + 10752864*x^3 + 24647840*x^2 + 24462336*x + 12648512)
 
gp: K = bnfinit(x^15 - 7*x^14 - 120*x^13 + 454*x^12 + 3979*x^11 + 327*x^10 - 38432*x^9 - 97106*x^8 - 228520*x^7 - 1075224*x^6 - 2667392*x^5 - 574336*x^4 + 10752864*x^3 + 24647840*x^2 + 24462336*x + 12648512, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 120 x^{13} + 454 x^{12} + 3979 x^{11} + 327 x^{10} - 38432 x^{9} - 97106 x^{8} - 228520 x^{7} - 1075224 x^{6} - 2667392 x^{5} - 574336 x^{4} + 10752864 x^{3} + 24647840 x^{2} + 24462336 x + 12648512 \)

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:  \(10467392050855906104242553244157476864=2^{24}\cdot 11^{6}\cdot 79^{5}\cdot 10698269^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $293.76$
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:  $2, 11, 79, 10698269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{305251864653769407147753176386714398184096} a^{14} - \frac{596909387258692598244267138063120910073}{305251864653769407147753176386714398184096} a^{13} + \frac{3014532020334219257503307732830733873955}{152625932326884703573876588193357199092048} a^{12} - \frac{4169455673582616623312036396721469457075}{152625932326884703573876588193357199092048} a^{11} + \frac{33857406045227314817545566063548843738075}{305251864653769407147753176386714398184096} a^{10} + \frac{6358119164226738083912247373362113987649}{305251864653769407147753176386714398184096} a^{9} + \frac{13026946080859200477860196853356892325205}{152625932326884703573876588193357199092048} a^{8} + \frac{33463694104663243659426995424552256629133}{152625932326884703573876588193357199092048} a^{7} - \frac{177730832232905715622434924025784609048}{9539120770430293973367286762084824943253} a^{6} - \frac{4409795144301224167566861284957812473787}{38156483081721175893469147048339299773012} a^{5} - \frac{5556364764665003072519588933316089282991}{38156483081721175893469147048339299773012} a^{4} + \frac{797521192952322649402200437680985101781}{19078241540860587946734573524169649886506} a^{3} - \frac{6296138033945417909746523695972285754343}{19078241540860587946734573524169649886506} a^{2} + \frac{2941072280685049299313160068269852448474}{9539120770430293973367286762084824943253} a + \frac{2762308788172742243816241229271320144415}{9539120770430293973367286762084824943253}$

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:  \( 64260011244400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.3.316.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15$ $15$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ $15$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.18.75$x^{8} + 24 x^{4} + 208$$8$$1$$18$$C_4\wr C_2$$[2, 2, 3]^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.6.3$x^{8} - 11 x^{4} + 847$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
79Data not computed
10698269Data not computed