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Properties

Label 15.15.8449050199...2176.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 11^{12}\cdot 23^{2}\cdot 89^{6}$
Root discriminant $98.88$
Ramified primes $2, 11, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T71

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-898633, -3113004, 10868994, 20264426, 2316335, -7465131, -1552520, 1181513, 230773, -100506, -14598, 4716, 405, -111, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 111*x^13 + 405*x^12 + 4716*x^11 - 14598*x^10 - 100506*x^9 + 230773*x^8 + 1181513*x^7 - 1552520*x^6 - 7465131*x^5 + 2316335*x^4 + 20264426*x^3 + 10868994*x^2 - 3113004*x - 898633)
 
gp: K = bnfinit(x^15 - 4*x^14 - 111*x^13 + 405*x^12 + 4716*x^11 - 14598*x^10 - 100506*x^9 + 230773*x^8 + 1181513*x^7 - 1552520*x^6 - 7465131*x^5 + 2316335*x^4 + 20264426*x^3 + 10868994*x^2 - 3113004*x - 898633, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 111 x^{13} + 405 x^{12} + 4716 x^{11} - 14598 x^{10} - 100506 x^{9} + 230773 x^{8} + 1181513 x^{7} - 1552520 x^{6} - 7465131 x^{5} + 2316335 x^{4} + 20264426 x^{3} + 10868994 x^{2} - 3113004 x - 898633 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(844905019931659078085517362176=2^{10}\cdot 11^{12}\cdot 23^{2}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.88$
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:  $2, 11, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{48} a^{11} - \frac{1}{24} a^{10} - \frac{1}{48} a^{9} - \frac{1}{16} a^{8} - \frac{5}{48} a^{7} + \frac{7}{24} a^{6} + \frac{11}{48} a^{5} - \frac{1}{6} a^{4} - \frac{5}{12} a^{3} - \frac{1}{12} a^{2} - \frac{1}{24} a - \frac{13}{48}$, $\frac{1}{288} a^{12} - \frac{1}{288} a^{11} + \frac{1}{96} a^{10} + \frac{7}{144} a^{9} + \frac{7}{72} a^{8} + \frac{3}{32} a^{7} + \frac{85}{288} a^{6} + \frac{1}{96} a^{5} + \frac{19}{144} a^{4} + \frac{19}{48} a^{3} - \frac{1}{24} a^{2} + \frac{41}{96} a + \frac{137}{288}$, $\frac{1}{39744} a^{13} + \frac{1}{2208} a^{12} - \frac{125}{19872} a^{11} + \frac{179}{39744} a^{10} - \frac{139}{3312} a^{9} - \frac{2987}{39744} a^{8} - \frac{569}{2484} a^{7} + \frac{4481}{19872} a^{6} - \frac{15331}{39744} a^{5} + \frac{116}{621} a^{4} + \frac{179}{6624} a^{3} + \frac{2521}{13248} a^{2} + \frac{305}{864} a - \frac{113}{1728}$, $\frac{1}{8338647398845188776023975894848} a^{14} - \frac{51481358945375042440782355}{4169323699422594388011987947424} a^{13} + \frac{3680992645548723254657613799}{4169323699422594388011987947424} a^{12} - \frac{72317550907376117669579365745}{8338647398845188776023975894848} a^{11} - \frac{63208850512271510735566403383}{2084661849711297194005993973712} a^{10} - \frac{105025416191051175148362440087}{8338647398845188776023975894848} a^{9} + \frac{251749844358803138808575492831}{2084661849711297194005993973712} a^{8} - \frac{443203625893689730747324570157}{4169323699422594388011987947424} a^{7} + \frac{829316180652599288227935162199}{2779549132948396258674658631616} a^{6} - \frac{41222027541501859626356719201}{90637471726578138869825824944} a^{5} + \frac{1378459808452536173044461911845}{4169323699422594388011987947424} a^{4} + \frac{603522002446429985879629763465}{2779549132948396258674658631616} a^{3} - \frac{1453304561675295726464781228929}{4169323699422594388011987947424} a^{2} + \frac{169297264020896816730960878399}{362549886906312555479303299776} a - \frac{36442208074089636683162949581}{90637471726578138869825824944}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15372643736.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T71:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 19440
The 39 conjugacy class representatives for [1/2.S(3)^5]5
Character table for [1/2.S(3)^5]5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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
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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ $15$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $15$ $15$ $15$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$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$$[\ ]^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.3.2.1$x^{3} - 89$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
89.3.2.1$x^{3} - 89$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
89.3.2.1$x^{3} - 89$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$