↑

Properties

Label 15.1.70997798514...3248.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 683^{7}$
Root discriminant $33.37$
Ramified primes $2, 683$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{15}$ (as 15T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-260, -1724, -2676, 3451, 1321, -1149, -192, 255, -237, -28, 101, -11, -12, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 9*x^13 - 12*x^12 - 11*x^11 + 101*x^10 - 28*x^9 - 237*x^8 + 255*x^7 - 192*x^6 - 1149*x^5 + 1321*x^4 + 3451*x^3 - 2676*x^2 - 1724*x - 260)
 
gp: K = bnfinit(x^15 - 5*x^14 + 9*x^13 - 12*x^12 - 11*x^11 + 101*x^10 - 28*x^9 - 237*x^8 + 255*x^7 - 192*x^6 - 1149*x^5 + 1321*x^4 + 3451*x^3 - 2676*x^2 - 1724*x - 260, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 9 x^{13} - 12 x^{12} - 11 x^{11} + 101 x^{10} - 28 x^{9} - 237 x^{8} + 255 x^{7} - 192 x^{6} - 1149 x^{5} + 1321 x^{4} + 3451 x^{3} - 2676 x^{2} - 1724 x - 260 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-70997798514319584693248=-\,2^{10}\cdot 683^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.37$
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:  $2, 683$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{26} a^{11} - \frac{5}{13} a^{10} + \frac{4}{13} a^{9} + \frac{9}{26} a^{8} - \frac{6}{13} a^{6} + \frac{1}{26} a^{5} - \frac{5}{13} a^{4} + \frac{3}{26} a^{2} + \frac{1}{13} a$, $\frac{1}{26} a^{12} + \frac{6}{13} a^{10} + \frac{11}{26} a^{9} + \frac{6}{13} a^{8} - \frac{6}{13} a^{7} + \frac{11}{26} a^{6} + \frac{2}{13} a^{4} + \frac{3}{26} a^{3} + \frac{3}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{5642} a^{13} + \frac{2}{403} a^{12} + \frac{3}{434} a^{11} - \frac{755}{5642} a^{10} + \frac{918}{2821} a^{9} + \frac{463}{5642} a^{8} + \frac{5}{434} a^{7} + \frac{577}{2821} a^{6} - \frac{2413}{5642} a^{5} - \frac{1325}{5642} a^{4} + \frac{129}{403} a^{3} - \frac{979}{5642} a^{2} - \frac{148}{2821} a + \frac{33}{217}$, $\frac{1}{6884933441052940} a^{14} + \frac{581655811}{245890480037605} a^{13} - \frac{105277546071337}{6884933441052940} a^{12} + \frac{108977963141727}{6884933441052940} a^{11} - \frac{148334749437905}{344246672052647} a^{10} - \frac{1963610368470169}{6884933441052940} a^{9} - \frac{51531693708949}{105922052939276} a^{8} - \frac{1381633942549711}{3442466720526470} a^{7} + \frac{1633767540653579}{6884933441052940} a^{6} + \frac{20511885768715}{80999216953564} a^{5} - \frac{192617773954131}{491780960075210} a^{4} + \frac{691585522898509}{6884933441052940} a^{3} + \frac{191291402283047}{1721233360263235} a^{2} + \frac{100792869058897}{1721233360263235} a - \frac{657338475341}{3782930462117}$

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:  $7$
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:  \( 993869.585586 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2732.1, 5.1.466489.1

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

Sibling fields

Galois closure: 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 $15$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ $15$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
683Data not computed