↑

Properties

Label 15.1.66959890291...5759.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3^{20}\cdot 79^{7}$
Root discriminant $33.24$
Ramified primes $3, 79$
Class number $2$
Class group $[2]$
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![1071, 1116, -297, 319, 93, 585, -556, 648, -126, 59, -39, 36, -19, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071)
 
gp: K = bnfinit(x^15 - 19*x^12 + 36*x^11 - 39*x^10 + 59*x^9 - 126*x^8 + 648*x^7 - 556*x^6 + 585*x^5 + 93*x^4 + 319*x^3 - 297*x^2 + 1116*x + 1071, 1)
 

Normalized defining polynomial

\( x^{15} - 19 x^{12} + 36 x^{11} - 39 x^{10} + 59 x^{9} - 126 x^{8} + 648 x^{7} - 556 x^{6} + 585 x^{5} + 93 x^{4} + 319 x^{3} - 297 x^{2} + 1116 x + 1071 \)

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:  \(-66959890291162926105759=-\,3^{20}\cdot 79^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.24$
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:  $3, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{12} - \frac{2}{9} a^{10} + \frac{4}{27} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{11}{27} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{27} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{27} a^{13} + \frac{1}{9} a^{11} + \frac{4}{27} a^{10} - \frac{1}{3} a^{9} - \frac{1}{9} a^{8} - \frac{11}{27} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{1}{27} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{2471757368730712177563} a^{14} - \frac{4684422197833498598}{823919122910237392521} a^{13} + \frac{34193401714778201344}{2471757368730712177563} a^{12} + \frac{155876107038501157024}{2471757368730712177563} a^{11} - \frac{377505485917664020561}{823919122910237392521} a^{10} + \frac{36077684463732543220}{353108195532958882509} a^{9} + \frac{813154968242241157375}{2471757368730712177563} a^{8} + \frac{122877994903923448642}{274639707636745797507} a^{7} - \frac{14679494588026834655}{353108195532958882509} a^{6} - \frac{750714920580290865272}{2471757368730712177563} a^{5} + \frac{22486512531327457374}{91546569212248599169} a^{4} - \frac{1088451758957536500293}{2471757368730712177563} a^{3} - \frac{899166207243851811}{13078081316035514167} a^{2} + \frac{71455685268025611289}{274639707636745797507} a + \frac{183374588726018816}{39234243948106542501}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 145411.427122 \)
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

3.1.6399.1, 5.1.6241.1

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 $15$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$