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Properties

Label 15.15.2131948569...6961.1
Degree $15$
Signature $[15, 0]$
Discriminant $31^{4}\cdot 61^{4}\cdot 401^{7}$
Root discriminant $122.63$
Ramified primes $31, 61, 401$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T35

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![478377, -97419, -3016619, -81156, 3008921, -230765, -1168291, 202802, 191854, -46097, -12252, 3454, 324, -102, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 102*x^13 + 324*x^12 + 3454*x^11 - 12252*x^10 - 46097*x^9 + 191854*x^8 + 202802*x^7 - 1168291*x^6 - 230765*x^5 + 3008921*x^4 - 81156*x^3 - 3016619*x^2 - 97419*x + 478377)
 
gp: K = bnfinit(x^15 - 3*x^14 - 102*x^13 + 324*x^12 + 3454*x^11 - 12252*x^10 - 46097*x^9 + 191854*x^8 + 202802*x^7 - 1168291*x^6 - 230765*x^5 + 3008921*x^4 - 81156*x^3 - 3016619*x^2 - 97419*x + 478377, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 102 x^{13} + 324 x^{12} + 3454 x^{11} - 12252 x^{10} - 46097 x^{9} + 191854 x^{8} + 202802 x^{7} - 1168291 x^{6} - 230765 x^{5} + 3008921 x^{4} - 81156 x^{3} - 3016619 x^{2} - 97419 x + 478377 \)

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:  \(21319485696611637812225259136961=31^{4}\cdot 61^{4}\cdot 401^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.63$
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:  $31, 61, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{2978716182416081630532266740084138941} a^{14} + \frac{38432054600296147085367575012354045}{2978716182416081630532266740084138941} a^{13} + \frac{386897082210895821346512975563189242}{2978716182416081630532266740084138941} a^{12} + \frac{371223920272318031484913983356884862}{2978716182416081630532266740084138941} a^{11} - \frac{1376312176619518303522842268413734203}{2978716182416081630532266740084138941} a^{10} + \frac{178580596686639381991951726669196512}{2978716182416081630532266740084138941} a^{9} - \frac{226259435583748631800240317961124017}{992905394138693876844088913361379647} a^{8} + \frac{142059909750771846500572810275627604}{2978716182416081630532266740084138941} a^{7} - \frac{1136607695034504130797564865259455235}{2978716182416081630532266740084138941} a^{6} - \frac{399548285057225599352053445189116733}{2978716182416081630532266740084138941} a^{5} + \frac{315940676404359301318174622777751137}{992905394138693876844088913361379647} a^{4} + \frac{382510645405452721006869760419143864}{2978716182416081630532266740084138941} a^{3} + \frac{1172311947504758033596173947532075784}{2978716182416081630532266740084138941} a^{2} - \frac{2872301123447278737361471265548907}{330968464712897958948029637787126549} a + \frac{146854638178752626699068166836301102}{330968464712897958948029637787126549}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 50365008603.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T35:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 810
The 24 conjugacy class representatives for 1/2[3^4:2]D(5)
Character table for 1/2[3^4:2]D(5) is not computed

Intermediate fields

5.5.160801.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.4.2$x^{6} - 31 x^{3} + 11532$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.6.4.3$x^{6} + 183 x^{3} + 14884$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
401Data not computed