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Properties

Label 15.1.34740892637...4263.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,7^{10}\cdot 103^{7}$
Root discriminant $31.82$
Ramified primes $7, 103$
Class number $3$
Class group $[3]$
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-225, 570, -773, 2129, -1080, 2102, -96, 334, 8, 113, -92, 27, 8, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225)
 
gp: K = bnfinit(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} + 3 x^{13} + 8 x^{12} + 27 x^{11} - 92 x^{10} + 113 x^{9} + 8 x^{8} + 334 x^{7} - 96 x^{6} + 2102 x^{5} - 1080 x^{4} + 2129 x^{3} - 773 x^{2} + 570 x - 225 \)

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:  \(-34740892637448264404263=-\,7^{10}\cdot 103^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.82$
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:  $7, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{15} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{15} a$, $\frac{1}{15} a^{10} - \frac{2}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{6} - \frac{2}{15} a^{5} - \frac{2}{15} a^{4} - \frac{1}{3} a^{3} - \frac{2}{5} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{6} - \frac{2}{15} a^{5} + \frac{1}{15} a^{4} - \frac{1}{3} a^{2} - \frac{2}{15} a$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{11} + \frac{1}{45} a^{9} - \frac{1}{9} a^{8} - \frac{11}{45} a^{7} - \frac{2}{45} a^{6} - \frac{2}{45} a^{5} - \frac{7}{15} a^{4} + \frac{1}{9} a^{3} + \frac{13}{45} a^{2} + \frac{4}{15} a$, $\frac{1}{458865} a^{13} - \frac{301}{152955} a^{12} + \frac{1406}{458865} a^{11} + \frac{8716}{458865} a^{10} + \frac{4652}{458865} a^{9} - \frac{76432}{458865} a^{8} + \frac{56456}{458865} a^{7} - \frac{167194}{458865} a^{6} + \frac{43664}{91773} a^{5} + \frac{16843}{41715} a^{4} - \frac{4}{99} a^{3} + \frac{104461}{458865} a^{2} + \frac{72163}{152955} a + \frac{1906}{10197}$, $\frac{1}{2085223431555} a^{14} + \frac{293780}{417044686311} a^{13} + \frac{2662008550}{417044686311} a^{12} + \frac{825215059}{695074477185} a^{11} - \frac{1662178274}{139014895437} a^{10} + \frac{10776400004}{417044686311} a^{9} - \frac{235308679919}{2085223431555} a^{8} - \frac{664871437094}{2085223431555} a^{7} + \frac{3057969838}{12637717767} a^{6} - \frac{8830938992}{46338298479} a^{5} + \frac{45854422466}{2085223431555} a^{4} - \frac{192370752688}{417044686311} a^{3} - \frac{836694004739}{2085223431555} a^{2} + \frac{202831342741}{695074477185} a - \frac{8758935170}{46338298479}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 147159.120328 \)
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

3.1.5047.1, 5.1.10609.1

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 $15$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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
7Data not computed
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$