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Properties

Label 15.3.23623665651...0321.2
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 19^{4}\cdot 151^{4}$
Root discriminant $36.16$
Ramified primes $3, 19, 151$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T89

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12377, -67569, -53802, -1765, 2940, 3636, -2828, -3204, -18, 110, 225, 18, -12, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 3*x^13 - 12*x^12 + 18*x^11 + 225*x^10 + 110*x^9 - 18*x^8 - 3204*x^7 - 2828*x^6 + 3636*x^5 + 2940*x^4 - 1765*x^3 - 53802*x^2 - 67569*x + 12377)
 
gp: K = bnfinit(x^15 + 3*x^13 - 12*x^12 + 18*x^11 + 225*x^10 + 110*x^9 - 18*x^8 - 3204*x^7 - 2828*x^6 + 3636*x^5 + 2940*x^4 - 1765*x^3 - 53802*x^2 - 67569*x + 12377, 1)
 

Normalized defining polynomial

\( x^{15} + 3 x^{13} - 12 x^{12} + 18 x^{11} + 225 x^{10} + 110 x^{9} - 18 x^{8} - 3204 x^{7} - 2828 x^{6} + 3636 x^{5} + 2940 x^{4} - 1765 x^{3} - 53802 x^{2} - 67569 x + 12377 \)

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:  \(236236656513512990640321=3^{20}\cdot 19^{4}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.16$
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:  $3, 19, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{27513854643664044798058129333984144} a^{14} - \frac{601370300933264411443064707936839}{27513854643664044798058129333984144} a^{13} + \frac{1010030989950815973243057922929117}{6878463660916011199514532333496036} a^{12} + \frac{1420207917787960701348734857071349}{3439231830458005599757266166748018} a^{11} - \frac{6232167497980294127470876357620267}{13756927321832022399029064666992072} a^{10} + \frac{6950698355110636809348356382895083}{27513854643664044798058129333984144} a^{9} - \frac{4204439888004691251451740440741007}{27513854643664044798058129333984144} a^{8} + \frac{13530330872392317129692390269968087}{27513854643664044798058129333984144} a^{7} + \frac{12937497173474140495802917089174315}{27513854643664044798058129333984144} a^{6} + \frac{11733147011110527616676191221018871}{27513854643664044798058129333984144} a^{5} - \frac{12244205768684541533403122453302317}{27513854643664044798058129333984144} a^{4} + \frac{855296405649048454712626284510359}{27513854643664044798058129333984144} a^{3} - \frac{6330801823653084941942393826373115}{13756927321832022399029064666992072} a^{2} - \frac{759941053380157767021685485138034}{1719615915229002799878633083374009} a + \frac{11109662190411744526822893820568175}{27513854643664044798058129333984144}$

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:  $8$
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:  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:  \( 673922.436733 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T89:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 60 conjugacy class representatives for 1/2[S(3)^5]S(5) are not computed
Character table for 1/2[S(3)^5]S(5) is not computed

Intermediate fields

5.1.2869.1

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
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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
151.4.3.1$x^{4} + 755$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$