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Properties

Label 15.3.62598702400...7049.2
Degree $15$
Signature $[3, 6]$
Discriminant $3^{24}\cdot 53^{6}$
Root discriminant $28.39$
Ramified primes $3, 53$
Class number $1$
Class group Trivial
Galois group $A_7$ (as 15T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, -351, -108, 1228, -564, -1419, 1517, 132, -957, 475, 87, -174, 59, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 3*x^13 + 59*x^12 - 174*x^11 + 87*x^10 + 475*x^9 - 957*x^8 + 132*x^7 + 1517*x^6 - 1419*x^5 - 564*x^4 + 1228*x^3 - 108*x^2 - 351*x + 29)
 
gp: K = bnfinit(x^15 - 6*x^14 + 3*x^13 + 59*x^12 - 174*x^11 + 87*x^10 + 475*x^9 - 957*x^8 + 132*x^7 + 1517*x^6 - 1419*x^5 - 564*x^4 + 1228*x^3 - 108*x^2 - 351*x + 29, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 3 x^{13} + 59 x^{12} - 174 x^{11} + 87 x^{10} + 475 x^{9} - 957 x^{8} + 132 x^{7} + 1517 x^{6} - 1419 x^{5} - 564 x^{4} + 1228 x^{3} - 108 x^{2} - 351 x + 29 \)

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:  \(6259870240060963847049=3^{24}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.39$
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:  $3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{4}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a$, $\frac{1}{448229805273} a^{14} - \frac{7065418846}{448229805273} a^{13} - \frac{16168000868}{448229805273} a^{12} - \frac{25794020543}{448229805273} a^{11} - \frac{152745714661}{448229805273} a^{10} + \frac{96602769760}{448229805273} a^{9} + \frac{33103486183}{149409935091} a^{8} + \frac{50680323232}{149409935091} a^{7} - \frac{37683319583}{149409935091} a^{6} + \frac{101532507347}{448229805273} a^{5} - \frac{61118992172}{448229805273} a^{4} + \frac{22373707061}{448229805273} a^{3} - \frac{65420059336}{448229805273} a^{2} - \frac{203126471969}{448229805273} a + \frac{32495992436}{448229805273}$

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

Class group and class number

Trivial group, which has 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:  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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 323534.535743 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_7$ (as 15T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2520
The 9 conjugacy class representatives for $A_7$
Character table for $A_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 7 sibling: 7.3.18429849.1
Degree 21 sibling: 21.5.115368463283917314495572165601.1
Degree 35 sibling: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$3$3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.9.16.4$x^{9} + 3 x^{8} + 3$$9$$1$$16$$C_3^2:C_4$$[2, 2]^{4}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$