Properties

Label 15.3.39671359416303616.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 73^{6}$
Root discriminant $12.78$
Ramified primes $2, 73$
Class number $1$
Class group Trivial
Galois Group $A_5$ (as 15T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 31, -83, 171, -282, 371, -383, 296, -150, 18, 46, -47, 24, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*x + 1)
gp: K = bnfinit(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 7 x^{14} \) \(\mathstrut +\mathstrut 24 x^{13} \) \(\mathstrut -\mathstrut 47 x^{12} \) \(\mathstrut +\mathstrut 46 x^{11} \) \(\mathstrut +\mathstrut 18 x^{10} \) \(\mathstrut -\mathstrut 150 x^{9} \) \(\mathstrut +\mathstrut 296 x^{8} \) \(\mathstrut -\mathstrut 383 x^{7} \) \(\mathstrut +\mathstrut 371 x^{6} \) \(\mathstrut -\mathstrut 282 x^{5} \) \(\mathstrut +\mathstrut 171 x^{4} \) \(\mathstrut -\mathstrut 83 x^{3} \) \(\mathstrut +\mathstrut 31 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(39671359416303616=2^{18}\cdot 73^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.78$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 73$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( a^{14} - 6 a^{13} + 18 a^{12} - 29 a^{11} + 17 a^{10} + 35 a^{9} - 115 a^{8} + 181 a^{7} - 202 a^{6} + 169 a^{5} - 113 a^{4} + 58 a^{3} - 25 a^{2} + 6 a - 1 \),  \( a - 1 \),  \( \frac{5}{4} a^{14} - 11 a^{13} + \frac{173}{4} a^{12} - \frac{387}{4} a^{11} + 113 a^{10} + \frac{21}{4} a^{9} - \frac{1149}{4} a^{8} + \frac{2475}{4} a^{7} - \frac{3259}{4} a^{6} + \frac{1575}{2} a^{5} - \frac{1163}{2} a^{4} + \frac{1351}{4} a^{3} - 152 a^{2} + 51 a - \frac{39}{4} \),  \( \frac{7}{4} a^{14} - \frac{41}{4} a^{13} + 30 a^{12} - \frac{93}{2} a^{11} + \frac{95}{4} a^{10} + \frac{249}{4} a^{9} - \frac{755}{4} a^{8} + \frac{577}{2} a^{7} - \frac{1273}{4} a^{6} + \frac{529}{2} a^{5} - \frac{357}{2} a^{4} + \frac{381}{4} a^{3} - \frac{169}{4} a^{2} + \frac{49}{4} a - \frac{9}{4} \),  \( a^{14} - 5 a^{13} + \frac{51}{4} a^{12} - \frac{63}{4} a^{11} + \frac{7}{4} a^{10} + \frac{127}{4} a^{9} - \frac{149}{2} a^{8} + 103 a^{7} - \frac{439}{4} a^{6} + 85 a^{5} - 56 a^{4} + 28 a^{3} - 12 a^{2} + \frac{19}{4} a - \frac{7}{4} \),  \( \frac{1}{2} a^{14} - \frac{7}{2} a^{13} + \frac{41}{4} a^{12} - \frac{55}{4} a^{11} - \frac{13}{4} a^{10} + \frac{173}{4} a^{9} - \frac{155}{2} a^{8} + 73 a^{7} - \frac{115}{4} a^{6} - 21 a^{5} + \frac{105}{2} a^{4} - \frac{95}{2} a^{3} + 30 a^{2} - \frac{55}{4} a + \frac{17}{4} \),  \( \frac{21}{4} a^{14} - \frac{65}{2} a^{13} + \frac{197}{2} a^{12} - 160 a^{11} + \frac{367}{4} a^{10} + 199 a^{9} - \frac{2553}{4} a^{8} + \frac{3955}{4} a^{7} - \frac{2159}{2} a^{6} + \frac{1773}{2} a^{5} - \frac{1143}{2} a^{4} + \frac{1169}{4} a^{3} - \frac{239}{2} a^{2} + \frac{139}{4} a - 6 \),  \( a^{14} - 6 a^{13} + 17 a^{12} - 24 a^{11} + 5 a^{10} + 47 a^{9} - 108 a^{8} + 141 a^{7} - 134 a^{6} + 96 a^{5} - 52 a^{4} + 23 a^{3} - 8 a^{2} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 170.85154758 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$A_5$ (as 15T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

5.1.341056.1

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

Sibling fields

Degree 5 sibling: 5.1.341056.1
Degree 6 sibling: 6.2.341056.1
Degree 10 sibling: 10.2.116319195136.1
Degree 12 sibling: Deg 12
Degree 20 sibling: Deg 20
Degree 30 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$