Properties

Label 15.3.126064044311049216.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{5}\cdot 47^{7}$
Root discriminant $13.81$
Ramified primes $2, 3, 47$
Class number $1$
Class group Trivial
Galois Group $D_5\times S_3$ (as 15T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 26, -62, 86, -67, 0, 83, -111, 52, 23, -40, 15, 3, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1)
gp: K = bnfinit(x^15 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 4 x^{14} \) \(\mathstrut +\mathstrut 3 x^{13} \) \(\mathstrut +\mathstrut 15 x^{12} \) \(\mathstrut -\mathstrut 40 x^{11} \) \(\mathstrut +\mathstrut 23 x^{10} \) \(\mathstrut +\mathstrut 52 x^{9} \) \(\mathstrut -\mathstrut 111 x^{8} \) \(\mathstrut +\mathstrut 83 x^{7} \) \(\mathstrut -\mathstrut 67 x^{5} \) \(\mathstrut +\mathstrut 86 x^{4} \) \(\mathstrut -\mathstrut 62 x^{3} \) \(\mathstrut +\mathstrut 26 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(126064044311049216=2^{10}\cdot 3^{5}\cdot 47^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.81$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{404281} a^{14} + \frac{182213}{404281} a^{13} - \frac{79463}{404281} a^{12} - \frac{185441}{404281} a^{11} + \frac{111805}{404281} a^{10} - \frac{160725}{404281} a^{9} + \frac{96929}{404281} a^{8} - \frac{116846}{404281} a^{7} + \frac{131366}{404281} a^{6} + \frac{44693}{404281} a^{5} - \frac{12150}{404281} a^{4} - \frac{93708}{404281} a^{3} + \frac{21618}{404281} a^{2} - \frac{1348}{3709} a + \frac{974}{404281}$

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:  \( \frac{576707}{404281} a^{14} - \frac{2056301}{404281} a^{13} + \frac{808695}{404281} a^{12} + \frac{9032887}{404281} a^{11} - \frac{19051762}{404281} a^{10} + \frac{4740791}{404281} a^{9} + \frac{32041413}{404281} a^{8} - \frac{49671324}{404281} a^{7} + \frac{26136313}{404281} a^{6} + \frac{11150664}{404281} a^{5} - \frac{33951362}{404281} a^{4} + \frac{35375566}{404281} a^{3} - \frac{20583883}{404281} a^{2} + \frac{53581}{3709} a - \frac{1046534}{404281} \),  \( \frac{188233}{404281} a^{14} - \frac{700411}{404281} a^{13} + \frac{433840}{404281} a^{12} + \frac{2740035}{404281} a^{11} - \frac{6733948}{404281} a^{10} + \frac{3449677}{404281} a^{9} + \frac{8929109}{404281} a^{8} - \frac{18366520}{404281} a^{7} + \frac{14123029}{404281} a^{6} - \frac{794422}{404281} a^{5} - \frac{11333201}{404281} a^{4} + \frac{15204744}{404281} a^{3} - \frac{11187139}{404281} a^{2} + \frac{42823}{3709} a - \frac{1013194}{404281} \),  \( a \),  \( a - 1 \),  \( \frac{409864}{404281} a^{14} - \frac{1492941}{404281} a^{13} + \frac{662890}{404281} a^{12} + \frac{6515034}{404281} a^{11} - \frac{14152384}{404281} a^{10} + \frac{3814674}{404281} a^{9} + \frac{24079208}{404281} a^{8} - \frac{37439817}{404281} a^{7} + \frac{18243289}{404281} a^{6} + \frac{10590948}{404281} a^{5} - \frac{25383945}{404281} a^{4} + \frac{24628991}{404281} a^{3} - \frac{13932279}{404281} a^{2} + \frac{29349}{3709} a - \frac{222092}{404281} \),  \( \frac{228992}{404281} a^{14} - \frac{926995}{404281} a^{13} + \frac{700795}{404281} a^{12} + \frac{3596454}{404281} a^{11} - \frac{9559352}{404281} a^{10} + \frac{5045850}{404281} a^{9} + \frac{13875660}{404281} a^{8} - \frac{27356636}{404281} a^{7} + \frac{17002226}{404281} a^{6} + \frac{5221594}{404281} a^{5} - \frac{18587884}{404281} a^{4} + \frac{18641508}{404281} a^{3} - \frac{11391657}{404281} a^{2} + \frac{29981}{3709} a - \frac{124904}{404281} \),  \( \frac{149248}{404281} a^{14} - \frac{652765}{404281} a^{13} + \frac{697873}{404281} a^{12} + \frac{1996016}{404281} a^{11} - \frac{6898412}{404281} a^{10} + \frac{6192550}{404281} a^{9} + \frac{6136584}{404281} a^{8} - \frac{20584923}{404281} a^{7} + \frac{20315442}{404281} a^{6} - \frac{4338446}{404281} a^{5} - \frac{11887064}{404281} a^{4} + \frac{17753694}{404281} a^{3} - \frac{14273232}{404281} a^{2} + \frac{60327}{3709} a - \frac{1386451}{404281} \),  \( \frac{114322}{404281} a^{14} - \frac{432501}{404281} a^{13} + \frac{229265}{404281} a^{12} + \frac{1722481}{404281} a^{11} - \frac{4019696}{404281} a^{10} + \frac{1785124}{404281} a^{9} + \frac{5434862}{404281} a^{8} - \frac{10731197}{404281} a^{7} + \frac{8687446}{404281} a^{6} - \frac{718694}{404281} a^{5} - \frac{7584123}{404281} a^{4} + \frac{10263268}{404281} a^{3} - \frac{6425253}{404281} a^{2} + \frac{25148}{3709} a - \frac{636209}{404281} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 280.863045663 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_3\times D_5$ (as 15T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 60
The 12 conjugacy class representatives for $D_5\times S_3$
Character table for $D_5\times S_3$

Intermediate fields

3.3.564.1, 5.1.2209.1

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

Sibling fields

Degree 30 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
2Data not computed
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
47Data not computed