Properties

Label 15.1.6401597515801839.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3^{9}\cdot 13^{3}\cdot 23^{6}$
Root discriminant $11.32$
Ramified primes $3, 13, 23$
Class number $1$
Class group Trivial
Galois Group $S_5 \times S_3$ (as 15T29)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, 11, 7, -14, -2, 12, 3, 3, -1, -4, 2, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 1)
gp: K = bnfinit(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut 2 x^{12} \) \(\mathstrut -\mathstrut 4 x^{11} \) \(\mathstrut -\mathstrut x^{10} \) \(\mathstrut +\mathstrut 3 x^{9} \) \(\mathstrut +\mathstrut 3 x^{8} \) \(\mathstrut +\mathstrut 12 x^{7} \) \(\mathstrut -\mathstrut 2 x^{6} \) \(\mathstrut -\mathstrut 14 x^{5} \) \(\mathstrut +\mathstrut 7 x^{4} \) \(\mathstrut +\mathstrut 11 x^{3} \) \(\mathstrut -\mathstrut 6 x^{2} \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-6401597515801839=-\,3^{9}\cdot 13^{3}\cdot 23^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.32$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} - \frac{7}{23} a^{11} - \frac{5}{23} a^{10} + \frac{10}{23} a^{9} - \frac{7}{23} a^{8} + \frac{5}{23} a^{7} + \frac{4}{23} a^{6} + \frac{9}{23} a^{5} + \frac{6}{23} a^{4} + \frac{7}{23} a^{3} + \frac{1}{23} a^{2} - \frac{7}{23} a + \frac{4}{23}$, $\frac{1}{1917533} a^{14} + \frac{29448}{1917533} a^{13} - \frac{761330}{1917533} a^{12} - \frac{611332}{1917533} a^{11} - \frac{232445}{1917533} a^{10} + \frac{820230}{1917533} a^{9} + \frac{707153}{1917533} a^{8} - \frac{293390}{1917533} a^{7} - \frac{138626}{1917533} a^{6} + \frac{781020}{1917533} a^{5} - \frac{300256}{1917533} a^{4} - \frac{327532}{1917533} a^{3} + \frac{534843}{1917533} a^{2} + \frac{642407}{1917533} a + \frac{946988}{1917533}$

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:  $7$
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:  \( \frac{360497}{1917533} a^{14} + \frac{286226}{1917533} a^{13} - \frac{682720}{1917533} a^{12} - \frac{117120}{1917533} a^{11} - \frac{616888}{1917533} a^{10} - \frac{2471842}{1917533} a^{9} - \frac{639983}{1917533} a^{8} + \frac{2154207}{1917533} a^{7} + \frac{7491088}{1917533} a^{6} + \frac{8248471}{1917533} a^{5} - \frac{1484900}{1917533} a^{4} - \frac{5293656}{1917533} a^{3} + \frac{3104612}{1917533} a^{2} + \frac{4585530}{1917533} a - \frac{404054}{1917533} \),  \( \frac{329265}{1917533} a^{14} - \frac{101693}{1917533} a^{13} - \frac{233360}{1917533} a^{12} + \frac{44452}{1917533} a^{11} - \frac{1008070}{1917533} a^{10} - \frac{987354}{1917533} a^{9} - \frac{380756}{1917533} a^{8} + \frac{1753964}{1917533} a^{7} + \frac{4851047}{1917533} a^{6} + \frac{4367316}{1917533} a^{5} - \frac{1376217}{1917533} a^{4} - \frac{1934350}{1917533} a^{3} + \frac{1434176}{1917533} a^{2} + \frac{159448}{1917533} a + \frac{713029}{1917533} \),  \( \frac{37196}{1917533} a^{14} + \frac{353094}{1917533} a^{13} - \frac{303336}{1917533} a^{12} - \frac{415161}{1917533} a^{11} + \frac{548932}{1917533} a^{10} - \frac{1426193}{1917533} a^{9} - \frac{871109}{1917533} a^{8} + \frac{1246541}{1917533} a^{7} + \frac{1497590}{1917533} a^{6} + \frac{5197230}{1917533} a^{5} + \frac{35101}{83371} a^{4} - \frac{5211786}{1917533} a^{3} + \frac{3367048}{1917533} a^{2} + \frac{3093189}{1917533} a - \frac{1249046}{1917533} \),  \( \frac{197086}{1917533} a^{14} + \frac{166476}{1917533} a^{13} - \frac{527130}{1917533} a^{12} - \frac{127337}{1917533} a^{11} + \frac{209004}{1917533} a^{10} - \frac{1936527}{1917533} a^{9} + \frac{322878}{1917533} a^{8} + \frac{1980237}{1917533} a^{7} + \frac{2767704}{1917533} a^{6} + \frac{4899196}{1917533} a^{5} - \frac{4353734}{1917533} a^{4} - \frac{4476132}{1917533} a^{3} + \frac{5946360}{1917533} a^{2} + \frac{2892370}{1917533} a - \frac{1996231}{1917533} \),  \( \frac{159678}{1917533} a^{14} - \frac{260140}{1917533} a^{13} + \frac{105394}{1917533} a^{12} + \frac{415045}{1917533} a^{11} - \frac{1084188}{1917533} a^{10} + \frac{18691}{83371} a^{9} - \frac{155327}{1917533} a^{8} - \frac{179471}{1917533} a^{7} + \frac{1645718}{1917533} a^{6} - \frac{1049807}{1917533} a^{5} + \frac{1550822}{1917533} a^{4} + \frac{4059235}{1917533} a^{3} - \frac{1091168}{1917533} a^{2} - \frac{3164245}{1917533} a + \frac{1499744}{1917533} \),  \( \frac{69054}{1917533} a^{14} - \frac{333353}{1917533} a^{13} + \frac{120441}{1917533} a^{12} + \frac{652890}{1917533} a^{11} - \frac{1005594}{1917533} a^{10} + \frac{989747}{1917533} a^{9} + \frac{40509}{83371} a^{8} - \frac{1517141}{1917533} a^{7} + \frac{395271}{1917533} a^{6} - \frac{3563031}{1917533} a^{5} - \frac{1344286}{1917533} a^{4} + \frac{6493316}{1917533} a^{3} + \frac{112377}{1917533} a^{2} - \frac{4104220}{1917533} a + \frac{431792}{1917533} \),  \( \frac{487045}{1917533} a^{14} - \frac{62083}{1917533} a^{13} - \frac{943508}{1917533} a^{12} + \frac{410055}{1917533} a^{11} - \frac{44659}{83371} a^{10} - \frac{2151367}{1917533} a^{9} + \frac{1228043}{1917533} a^{8} + \frac{3344595}{1917533} a^{7} + \frac{7323747}{1917533} a^{6} + \frac{4611865}{1917533} a^{5} - \frac{9367957}{1917533} a^{4} - \frac{4920090}{1917533} a^{3} + \frac{5922147}{1917533} a^{2} + \frac{2760191}{1917533} a - \frac{1860241}{1917533} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 46.8775403574 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_5 \times S_3$ (as 15T29):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$ is not computed

Intermediate fields

3.1.23.1, 5.1.8073.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 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 $15$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$