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Properties

Label 15.15.4973877463...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{20}\cdot 5^{15}\cdot 7^{6}\cdot 97$
Root discriminant $111.29$
Ramified primes $2, 3, 5, 7, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15101, 81765, 84420, -219405, -484425, -163251, 216410, 125715, -32760, -25260, 1860, 2070, -35, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 75*x^13 - 35*x^12 + 2070*x^11 + 1860*x^10 - 25260*x^9 - 32760*x^8 + 125715*x^7 + 216410*x^6 - 163251*x^5 - 484425*x^4 - 219405*x^3 + 84420*x^2 + 81765*x + 15101)
 
gp: K = bnfinit(x^15 - 75*x^13 - 35*x^12 + 2070*x^11 + 1860*x^10 - 25260*x^9 - 32760*x^8 + 125715*x^7 + 216410*x^6 - 163251*x^5 - 484425*x^4 - 219405*x^3 + 84420*x^2 + 81765*x + 15101, 1)
 

Normalized defining polynomial

\( x^{15} - 75 x^{13} - 35 x^{12} + 2070 x^{11} + 1860 x^{10} - 25260 x^{9} - 32760 x^{8} + 125715 x^{7} + 216410 x^{6} - 163251 x^{5} - 484425 x^{4} - 219405 x^{3} + 84420 x^{2} + 81765 x + 15101 \)

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:  \(4973877463168144125000000000000=2^{12}\cdot 3^{20}\cdot 5^{15}\cdot 7^{6}\cdot 97\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.29$
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:  $2, 3, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3309476181358659206539250013} a^{14} + \frac{95340175803522864283207354}{1103158727119553068846416671} a^{13} + \frac{23912699126763771795212675}{1103158727119553068846416671} a^{12} - \frac{1380222092394276416629850}{1103158727119553068846416671} a^{11} - \frac{180619306070925811967259040}{3309476181358659206539250013} a^{10} - \frac{41665938640600573683450053}{1103158727119553068846416671} a^{9} - \frac{930678383954026900095391268}{3309476181358659206539250013} a^{8} + \frac{216282397146569039987818679}{1103158727119553068846416671} a^{7} - \frac{272547605196970411649074640}{1103158727119553068846416671} a^{6} - \frac{664187098398882508931540747}{3309476181358659206539250013} a^{5} - \frac{682053960007301007494154535}{3309476181358659206539250013} a^{4} - \frac{512232541605219696687212567}{1103158727119553068846416671} a^{3} + \frac{416920859658313589392622106}{1103158727119553068846416671} a^{2} + \frac{833654731701815803216644371}{3309476181358659206539250013} a - \frac{462019864771343643810213521}{1103158727119553068846416671}$

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:  \( 63024421528.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 155520
The 63 conjugacy class representatives for [S(3)^5]F(5)=S(3)wrF(5) are not computed
Character table for [S(3)^5]F(5)=S(3)wrF(5) is not computed

Intermediate fields

5.5.2450000.1

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 R R R $15$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
2Data not computed
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.19$x^{12} + 66 x^{11} - 45 x^{10} - 120 x^{9} + 9 x^{8} + 108 x^{7} + 18 x^{6} - 108 x^{5} - 81 x^{4} - 54 x^{3} - 81 x^{2} - 81$$3$$4$$16$12T46$[2, 2]^{8}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed