Properties

Label 15.3.23713122135310336.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 67^{6}$
Root discriminant $12.35$
Ramified primes $2, 67$
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, 0, -9, 9, 12, -24, 0, 36, -31, -8, 29, -15, -5, 10, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 10*x^13 - 5*x^12 - 15*x^11 + 29*x^10 - 8*x^9 - 31*x^8 + 36*x^7 - 24*x^5 + 12*x^4 + 9*x^3 - 9*x^2 + 1)
gp: K = bnfinit(x^15 - 5*x^14 + 10*x^13 - 5*x^12 - 15*x^11 + 29*x^10 - 8*x^9 - 31*x^8 + 36*x^7 - 24*x^5 + 12*x^4 + 9*x^3 - 9*x^2 + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 5 x^{14} \) \(\mathstrut +\mathstrut 10 x^{13} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut -\mathstrut 15 x^{11} \) \(\mathstrut +\mathstrut 29 x^{10} \) \(\mathstrut -\mathstrut 8 x^{9} \) \(\mathstrut -\mathstrut 31 x^{8} \) \(\mathstrut +\mathstrut 36 x^{7} \) \(\mathstrut -\mathstrut 24 x^{5} \) \(\mathstrut +\mathstrut 12 x^{4} \) \(\mathstrut +\mathstrut 9 x^{3} \) \(\mathstrut -\mathstrut 9 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(23713122135310336=2^{18}\cdot 67^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.35$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2474} a^{14} + \frac{204}{1237} a^{13} + \frac{141}{1237} a^{12} - \frac{527}{1237} a^{11} + \frac{107}{2474} a^{10} - \frac{156}{1237} a^{9} - \frac{108}{1237} a^{8} + \frac{531}{1237} a^{7} + \frac{372}{1237} a^{6} + \frac{248}{1237} a^{5} - \frac{259}{1237} a^{4} - \frac{579}{1237} a^{3} - \frac{763}{2474} a^{2} - \frac{465}{1237} a - \frac{310}{1237}$

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:  \( \frac{694}{1237} a^{14} - \frac{2595}{1237} a^{13} + \frac{6709}{2474} a^{12} + \frac{2065}{1237} a^{11} - \frac{9858}{1237} a^{10} + \frac{7369}{1237} a^{9} + \frac{18101}{2474} a^{8} - \frac{16305}{1237} a^{7} + \frac{1744}{1237} a^{6} + \frac{13945}{1237} a^{5} - \frac{6947}{1237} a^{4} - \frac{5787}{1237} a^{3} + \frac{8573}{1237} a^{2} + \frac{1531}{1237} a - \frac{3319}{2474} \),  \( \frac{694}{1237} a^{14} - \frac{2595}{1237} a^{13} + \frac{6709}{2474} a^{12} + \frac{2065}{1237} a^{11} - \frac{9858}{1237} a^{10} + \frac{7369}{1237} a^{9} + \frac{18101}{2474} a^{8} - \frac{16305}{1237} a^{7} + \frac{1744}{1237} a^{6} + \frac{13945}{1237} a^{5} - \frac{6947}{1237} a^{4} - \frac{5787}{1237} a^{3} + \frac{8573}{1237} a^{2} + \frac{1531}{1237} a - \frac{5793}{2474} \),  \( \frac{1169}{2474} a^{14} - \frac{4241}{2474} a^{13} + \frac{4327}{2474} a^{12} + \frac{3674}{1237} a^{11} - \frac{20883}{2474} a^{10} + \frac{10083}{2474} a^{9} + \frac{25821}{2474} a^{8} - \frac{17553}{1237} a^{7} - \frac{1793}{1237} a^{6} + \frac{17772}{1237} a^{5} - \frac{7128}{1237} a^{4} - \frac{6397}{1237} a^{3} + \frac{13537}{2474} a^{2} + \frac{5101}{2474} a - \frac{3609}{2474} \),  \( \frac{1169}{2474} a^{14} - \frac{4241}{2474} a^{13} + \frac{4327}{2474} a^{12} + \frac{3674}{1237} a^{11} - \frac{20883}{2474} a^{10} + \frac{10083}{2474} a^{9} + \frac{25821}{2474} a^{8} - \frac{17553}{1237} a^{7} - \frac{1793}{1237} a^{6} + \frac{17772}{1237} a^{5} - \frac{7128}{1237} a^{4} - \frac{6397}{1237} a^{3} + \frac{13537}{2474} a^{2} + \frac{5101}{2474} a - \frac{6083}{2474} \),  \( \frac{73}{1237} a^{14} + \frac{96}{1237} a^{13} - \frac{2123}{2474} a^{12} + \frac{2226}{1237} a^{11} - \frac{848}{1237} a^{10} - \frac{2984}{1237} a^{9} + \frac{9285}{2474} a^{8} - \frac{405}{1237} a^{7} - \frac{3827}{1237} a^{6} + \frac{2809}{1237} a^{5} - \frac{704}{1237} a^{4} + \frac{819}{1237} a^{3} - \frac{34}{1237} a^{2} + \frac{145}{1237} a + \frac{2255}{2474} \),  \( \frac{802}{1237} a^{14} - \frac{7363}{2474} a^{13} + \frac{13193}{2474} a^{12} - \frac{1674}{1237} a^{11} - \frac{11909}{1237} a^{10} + \frac{35173}{2474} a^{9} + \frac{1133}{2474} a^{8} - \frac{22835}{1237} a^{7} + \frac{17772}{1237} a^{6} + \frac{6900}{1237} a^{5} - \frac{13411}{1237} a^{4} + \frac{1508}{1237} a^{3} + \frac{7811}{1237} a^{2} - \frac{6083}{2474} a - \frac{1169}{2474} \),  \( \frac{581}{2474} a^{14} - \frac{1465}{1237} a^{13} + \frac{2753}{1237} a^{12} - \frac{648}{1237} a^{11} - \frac{12053}{2474} a^{10} + \frac{9561}{1237} a^{9} - \frac{898}{1237} a^{8} - \frac{13109}{1237} a^{7} + \frac{13264}{1237} a^{6} + \frac{1833}{1237} a^{5} - \frac{11935}{1237} a^{4} + \frac{6250}{1237} a^{3} + \frac{6965}{2474} a^{2} - \frac{4210}{1237} a - \frac{745}{1237} \),  \( \frac{2773}{2474} a^{14} - \frac{5802}{1237} a^{13} + \frac{8760}{1237} a^{12} + \frac{2000}{1237} a^{11} - \frac{44701}{2474} a^{10} + \frac{22628}{1237} a^{9} + \frac{13477}{1237} a^{8} - \frac{40388}{1237} a^{7} + \frac{15979}{1237} a^{6} + \frac{24672}{1237} a^{5} - \frac{21776}{1237} a^{4} - \frac{3652}{1237} a^{3} + \frac{29159}{2474} a^{2} - \frac{1728}{1237} a - \frac{3626}{1237} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 133.707274982 \)
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

5.1.287296.2

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

Sibling fields

Degree 5 sibling: 5.1.287296.2
Degree 6 sibling: 6.2.287296.2
Degree 10 sibling: 10.2.82538991616.1
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$