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Properties

Label 15.3.17518045204...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 5^{18}\cdot 257^{5}$
Root discriminant $76.37$
Ramified primes $2, 5, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26944, -74240, 73840, 16840, -30800, 30184, -17380, 4090, -1320, 1625, -384, 135, -50, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 10*x^13 - 50*x^12 + 135*x^11 - 384*x^10 + 1625*x^9 - 1320*x^8 + 4090*x^7 - 17380*x^6 + 30184*x^5 - 30800*x^4 + 16840*x^3 + 73840*x^2 - 74240*x + 26944)
 
gp: K = bnfinit(x^15 - 10*x^13 - 50*x^12 + 135*x^11 - 384*x^10 + 1625*x^9 - 1320*x^8 + 4090*x^7 - 17380*x^6 + 30184*x^5 - 30800*x^4 + 16840*x^3 + 73840*x^2 - 74240*x + 26944, 1)
 

Normalized defining polynomial

\( x^{15} - 10 x^{13} - 50 x^{12} + 135 x^{11} - 384 x^{10} + 1625 x^{9} - 1320 x^{8} + 4090 x^{7} - 17380 x^{6} + 30184 x^{5} - 30800 x^{4} + 16840 x^{3} + 73840 x^{2} - 74240 x + 26944 \)

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:  \(17518045204015625000000000000=2^{12}\cdot 5^{18}\cdot 257^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.37$
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:  $2, 5, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{60} a^{10} + \frac{1}{30} a^{8} + \frac{7}{30} a^{7} - \frac{29}{60} a^{6} + \frac{7}{15} a^{5} - \frac{1}{4} a^{4} - \frac{2}{15} a^{3} - \frac{13}{30} a^{2} - \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{60} a^{11} + \frac{1}{30} a^{9} + \frac{1}{30} a^{8} + \frac{7}{60} a^{7} - \frac{2}{15} a^{6} - \frac{1}{20} a^{5} - \frac{2}{15} a^{4} + \frac{11}{30} a^{3} - \frac{7}{15} a^{2} + \frac{4}{15} a + \frac{1}{5}$, $\frac{1}{120} a^{12} + \frac{1}{60} a^{9} + \frac{1}{40} a^{8} - \frac{3}{10} a^{7} + \frac{11}{24} a^{6} - \frac{1}{30} a^{5} + \frac{13}{30} a^{4} + \frac{2}{5} a^{3} - \frac{13}{30} a^{2} - \frac{1}{3} a + \frac{2}{15}$, $\frac{1}{720} a^{13} - \frac{1}{120} a^{11} - \frac{1}{360} a^{10} - \frac{11}{240} a^{9} + \frac{1}{45} a^{8} + \frac{1}{144} a^{7} - \frac{37}{90} a^{6} + \frac{53}{120} a^{5} - \frac{3}{20} a^{4} - \frac{1}{9} a^{3} + \frac{7}{18} a^{2} - \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{75251192786398811269294560} a^{14} + \frac{10402553517520264747939}{18812798196599702817323640} a^{13} - \frac{9614272202128329356543}{4180621821466600626071920} a^{12} + \frac{118376531621072099411579}{37625596393199405634647280} a^{11} + \frac{240464492676217161466279}{75251192786398811269294560} a^{10} + \frac{51063895038869787287771}{3762559639319940563464728} a^{9} - \frac{315622787121276344241937}{25083730928799603756431520} a^{8} + \frac{71819884211254982354339}{6270932732199900939107880} a^{7} - \frac{16144906204750161846281389}{37625596393199405634647280} a^{6} + \frac{685378803990596844236221}{2090310910733300313035960} a^{5} - \frac{482788544353120859915981}{4703199549149925704330910} a^{4} - \frac{941876224611667493291486}{2351599774574962852165455} a^{3} + \frac{5177687563475980877251}{37776703206023499633180} a^{2} - \frac{739264709372173192993541}{4703199549149925704330910} a + \frac{689619283935132615854491}{2351599774574962852165455}$

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:  \( 504015205.308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(5)wrS(3) is not computed

Intermediate fields

3.3.257.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $15$ $15$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.12.14$x^{10} + 10 x^{7} + 10 x^{5} + 100 x^{4} + 50 x^{2} + 25$$5$$2$$12$$D_5^2$$[3/2, 3/2]_{2}^{2}$
257Data not computed