Properties

Label 15.3.69184806150882304.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 47^{4}\cdot 61^{4}$
Root discriminant $13.26$
Ramified primes $2, 47, 61$
Class number $1$
Class group Trivial
Galois Group 15T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 1, 3, 1, 1, 4, 7, 3, -2, -5, -3, -3, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + x^13 - 3*x^12 - 3*x^11 - 5*x^10 - 2*x^9 + 3*x^8 + 7*x^7 + 4*x^6 + x^5 + x^4 + 3*x^3 + x^2 - 1)
gp: K = bnfinit(x^15 + x^13 - 3*x^12 - 3*x^11 - 5*x^10 - 2*x^9 + 3*x^8 + 7*x^7 + 4*x^6 + x^5 + x^4 + 3*x^3 + x^2 - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut +\mathstrut x^{13} \) \(\mathstrut -\mathstrut 3 x^{12} \) \(\mathstrut -\mathstrut 3 x^{11} \) \(\mathstrut -\mathstrut 5 x^{10} \) \(\mathstrut -\mathstrut 2 x^{9} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut +\mathstrut 7 x^{7} \) \(\mathstrut +\mathstrut 4 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut +\mathstrut 3 x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut -\mathstrut 1 \)

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:  \(69184806150882304=2^{10}\cdot 47^{4}\cdot 61^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.26$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 47, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{92} a^{14} + \frac{2}{23} a^{13} - \frac{1}{23} a^{12} + \frac{11}{92} a^{11} - \frac{7}{92} a^{10} + \frac{2}{23} a^{9} + \frac{39}{92} a^{8} - \frac{7}{92} a^{7} + \frac{5}{23} a^{6} - \frac{43}{92} a^{5} - \frac{21}{92} a^{4} + \frac{10}{23} a^{3} - \frac{11}{46} a^{2} - \frac{37}{92} a + \frac{13}{46}$

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:  \( a \),  \( \frac{5}{46} a^{14} + \frac{11}{92} a^{13} + \frac{29}{92} a^{12} - \frac{51}{92} a^{11} - \frac{6}{23} a^{10} - \frac{173}{92} a^{9} - \frac{47}{92} a^{8} - \frac{6}{23} a^{7} + \frac{131}{92} a^{6} + \frac{237}{92} a^{5} + \frac{33}{46} a^{4} - \frac{37}{92} a^{3} + \frac{79}{92} a^{2} + \frac{21}{92} a + \frac{19}{23} \),  \( \frac{4}{23} a^{14} - \frac{5}{46} a^{13} + \frac{5}{92} a^{12} - \frac{77}{92} a^{11} + \frac{3}{92} a^{10} - \frac{14}{23} a^{9} + \frac{141}{92} a^{8} + \frac{95}{92} a^{7} + \frac{34}{23} a^{6} - \frac{67}{92} a^{5} - \frac{221}{92} a^{4} - \frac{25}{46} a^{3} + \frac{85}{92} a^{2} - \frac{63}{92} a - \frac{21}{92} \),  \( \frac{9}{92} a^{14} + \frac{13}{46} a^{13} + \frac{5}{46} a^{12} + \frac{7}{92} a^{11} - \frac{109}{92} a^{10} - \frac{28}{23} a^{9} - \frac{201}{92} a^{8} - \frac{17}{92} a^{7} + \frac{22}{23} a^{6} + \frac{257}{92} a^{5} + \frac{133}{92} a^{4} + \frac{21}{23} a^{3} + \frac{39}{46} a^{2} + \frac{127}{92} a + \frac{1}{23} \),  \( \frac{1}{23} a^{14} + \frac{9}{92} a^{13} + \frac{15}{46} a^{12} - \frac{1}{46} a^{11} - \frac{5}{92} a^{10} - \frac{129}{92} a^{9} - \frac{30}{23} a^{8} - \frac{189}{92} a^{7} + \frac{11}{92} a^{6} + \frac{26}{23} a^{5} + \frac{261}{92} a^{4} + \frac{137}{92} a^{3} + \frac{24}{23} a^{2} + \frac{41}{46} a + \frac{35}{92} \),  \( \frac{43}{92} a^{14} - \frac{1}{92} a^{13} + \frac{29}{46} a^{12} - \frac{125}{92} a^{11} - \frac{35}{23} a^{10} - \frac{231}{92} a^{9} - \frac{163}{92} a^{8} + \frac{34}{23} a^{7} + \frac{285}{92} a^{6} + \frac{267}{92} a^{5} + \frac{33}{23} a^{4} - \frac{5}{92} a^{3} - \frac{13}{46} a^{2} + \frac{19}{92} a + \frac{83}{92} \),  \( \frac{4}{23} a^{14} - \frac{33}{92} a^{13} + \frac{51}{92} a^{12} - \frac{77}{92} a^{11} + \frac{18}{23} a^{10} - \frac{79}{92} a^{9} + \frac{3}{92} a^{8} - \frac{5}{23} a^{7} - \frac{71}{92} a^{6} - \frac{21}{92} a^{5} + \frac{85}{46} a^{4} + \frac{65}{92} a^{3} + \frac{39}{92} a^{2} + \frac{29}{92} a + \frac{12}{23} \),  \( \frac{4}{23} a^{14} - \frac{5}{46} a^{13} + \frac{5}{92} a^{12} - \frac{77}{92} a^{11} + \frac{3}{92} a^{10} - \frac{14}{23} a^{9} + \frac{141}{92} a^{8} + \frac{95}{92} a^{7} + \frac{34}{23} a^{6} - \frac{67}{92} a^{5} - \frac{221}{92} a^{4} - \frac{25}{46} a^{3} - \frac{7}{92} a^{2} + \frac{29}{92} a - \frac{21}{92} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 273.665310463 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T28:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 11 conjugacy class representatives for S_6(15)
Character table for S_6(15)

Intermediate fields

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

Sibling fields

Degree 6 siblings: 6.0.45872.1, 6.4.94263393452.1
Degree 10 sibling: 10.4.1508214295232.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 sibling: Deg 15
Degree 20 siblings: Deg 20, 20.0.2274710360342158457933824.1, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed