Properties

Label 15.3.57352136505929721.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{18}\cdot 23^{6}$
Root discriminant $13.10$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois Group 15T53

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -24, 57, -69, 63, -81, 81, -63, 40, -33, 22, -10, 4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3)
gp: K = bnfinit(x^15 - 2*x^14 + 4*x^13 - 10*x^12 + 22*x^11 - 33*x^10 + 40*x^9 - 63*x^8 + 81*x^7 - 81*x^6 + 63*x^5 - 69*x^4 + 57*x^3 - 24*x^2 + 3, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut -\mathstrut 10 x^{12} \) \(\mathstrut +\mathstrut 22 x^{11} \) \(\mathstrut -\mathstrut 33 x^{10} \) \(\mathstrut +\mathstrut 40 x^{9} \) \(\mathstrut -\mathstrut 63 x^{8} \) \(\mathstrut +\mathstrut 81 x^{7} \) \(\mathstrut -\mathstrut 81 x^{6} \) \(\mathstrut +\mathstrut 63 x^{5} \) \(\mathstrut -\mathstrut 69 x^{4} \) \(\mathstrut +\mathstrut 57 x^{3} \) \(\mathstrut -\mathstrut 24 x^{2} \) \(\mathstrut +\mathstrut 3 \)

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:  \(57352136505929721=3^{18}\cdot 23^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.10$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{23465906} a^{14} + \frac{2240267}{23465906} a^{13} - \frac{2566391}{11732953} a^{12} - \frac{1104218}{11732953} a^{11} + \frac{2247013}{23465906} a^{10} - \frac{4321609}{23465906} a^{9} - \frac{3175301}{23465906} a^{8} - \frac{1626737}{11732953} a^{7} - \frac{5745389}{23465906} a^{6} - \frac{1583690}{11732953} a^{5} + \frac{1692465}{23465906} a^{4} + \frac{1356674}{11732953} a^{3} + \frac{4693738}{11732953} a^{2} - \frac{1004432}{11732953} a - \frac{5214056}{11732953}$

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:  \( \frac{2121397}{11732953} a^{14} - \frac{2901785}{23465906} a^{13} + \frac{4831572}{11732953} a^{12} - \frac{13105145}{11732953} a^{11} + \frac{24622992}{11732953} a^{10} - \frac{23950351}{11732953} a^{9} + \frac{25494957}{11732953} a^{8} - \frac{129824339}{23465906} a^{7} + \frac{80636135}{23465906} a^{6} - \frac{50612781}{23465906} a^{5} + \frac{959028}{11732953} a^{4} - \frac{51502832}{11732953} a^{3} - \frac{24094941}{11732953} a^{2} + \frac{55052445}{23465906} a - \frac{34040249}{23465906} \),  \( \frac{1377556}{11732953} a^{14} - \frac{1914232}{11732953} a^{13} + \frac{5223316}{11732953} a^{12} - \frac{25491045}{23465906} a^{11} + \frac{55824301}{23465906} a^{10} - \frac{40186075}{11732953} a^{9} + \frac{54195386}{11732953} a^{8} - \frac{190218161}{23465906} a^{7} + \frac{217764805}{23465906} a^{6} - \frac{231915293}{23465906} a^{5} + \frac{92355581}{11732953} a^{4} - \frac{249363269}{23465906} a^{3} + \frac{76813599}{11732953} a^{2} - \frac{58854279}{23465906} a + \frac{3148996}{11732953} \),  \( \frac{329141}{11732953} a^{14} + \frac{845771}{23465906} a^{13} + \frac{2605557}{23465906} a^{12} - \frac{2125487}{23465906} a^{11} + \frac{6559909}{23465906} a^{10} - \frac{6966593}{23465906} a^{9} + \frac{17282927}{23465906} a^{8} - \frac{20264383}{11732953} a^{7} + \frac{19504899}{23465906} a^{6} - \frac{30013577}{11732953} a^{5} + \frac{14212984}{11732953} a^{4} - \frac{38308292}{11732953} a^{3} + \frac{463284}{11732953} a^{2} - \frac{36543783}{23465906} a + \frac{3659969}{11732953} \),  \( \frac{1280919}{23465906} a^{14} + \frac{1852445}{23465906} a^{13} - \frac{221789}{11732953} a^{12} - \frac{931431}{23465906} a^{11} - \frac{2127171}{11732953} a^{10} + \frac{23078541}{23465906} a^{9} - \frac{28292357}{23465906} a^{8} + \frac{2646417}{23465906} a^{7} - \frac{29578768}{11732953} a^{6} + \frac{62726321}{23465906} a^{5} - \frac{57548193}{23465906} a^{4} + \frac{8470293}{23465906} a^{3} - \frac{42386474}{11732953} a^{2} + \frac{43987085}{23465906} a + \frac{637585}{11732953} \),  \( \frac{450695}{11732953} a^{14} - \frac{2134850}{11732953} a^{13} + \frac{5256557}{23465906} a^{12} - \frac{14121201}{23465906} a^{11} + \frac{34036445}{23465906} a^{10} - \frac{56075745}{23465906} a^{9} + \frac{57583007}{23465906} a^{8} - \frac{79457181}{23465906} a^{7} + \frac{68097651}{11732953} a^{6} - \frac{100403569}{23465906} a^{5} + \frac{37971598}{11732953} a^{4} - \frac{33314330}{11732953} a^{3} + \frac{56041738}{11732953} a^{2} + \frac{11823671}{11732953} a - \frac{22710401}{23465906} \),  \( \frac{232143}{11732953} a^{14} - \frac{839544}{11732953} a^{13} + \frac{630089}{11732953} a^{12} - \frac{1577013}{11732953} a^{11} + \frac{4714385}{11732953} a^{10} - \frac{5131822}{11732953} a^{9} - \frac{1127818}{11732953} a^{8} + \frac{2435734}{11732953} a^{7} + \frac{3326601}{11732953} a^{6} + \frac{7336217}{11732953} a^{5} - \frac{19227569}{11732953} a^{4} + \frac{12895912}{11732953} a^{3} - \frac{6650293}{11732953} a^{2} + \frac{17700292}{11732953} a + \frac{56662}{11732953} \),  \( \frac{2121397}{11732953} a^{14} - \frac{2901785}{23465906} a^{13} + \frac{4831572}{11732953} a^{12} - \frac{13105145}{11732953} a^{11} + \frac{24622992}{11732953} a^{10} - \frac{23950351}{11732953} a^{9} + \frac{25494957}{11732953} a^{8} - \frac{129824339}{23465906} a^{7} + \frac{80636135}{23465906} a^{6} - \frac{50612781}{23465906} a^{5} + \frac{959028}{11732953} a^{4} - \frac{51502832}{11732953} a^{3} - \frac{24094941}{11732953} a^{2} + \frac{78518351}{23465906} a - \frac{34040249}{23465906} \),  \( \frac{3933711}{11732953} a^{14} - \frac{10518349}{23465906} a^{13} + \frac{23363169}{23465906} a^{12} - \frac{60118519}{23465906} a^{11} + \frac{124026527}{23465906} a^{10} - \frac{158785645}{23465906} a^{9} + \frac{171123377}{23465906} a^{8} - \frac{155988674}{11732953} a^{7} + \frac{329598227}{23465906} a^{6} - \frac{147029232}{11732953} a^{5} + \frac{85598637}{11732953} a^{4} - \frac{139929873}{11732953} a^{3} + \frac{72240228}{11732953} a^{2} - \frac{23279689}{23465906} a - \frac{17951677}{11732953} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 279.393832589 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T53:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 4860
The 24 conjugacy class representatives for [3^4]A(5)
Character table for [3^4]A(5) is not computed

Intermediate fields

5.1.42849.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.9.15.23$x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 3$$9$$1$$15$$C_3 \wr S_3 $$[3/2, 3/2, 2]_{2}^{3}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.9.6.1$x^{9} - 529 x^{3} + 48668$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$