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Properties

Label 15.11.1556733003...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 71^{2}\cdot 337^{2}\cdot 2843^{4}\cdot 116719^{2}$
Root discriminant $2587.11$
Ramified primes $2, 5, 7, 13, 71, 337, 2843, 116719$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T98

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2843000, -29851500, 135184650, -344500525, 540397440, -536215885, 333271318, -124445806, 25958664, -2848725, 261104, -32503, -144, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 36*x^13 - 144*x^12 - 32503*x^11 + 261104*x^10 - 2848725*x^9 + 25958664*x^8 - 124445806*x^7 + 333271318*x^6 - 536215885*x^5 + 540397440*x^4 - 344500525*x^3 + 135184650*x^2 - 29851500*x + 2843000)
 
gp: K = bnfinit(x^15 + 36*x^13 - 144*x^12 - 32503*x^11 + 261104*x^10 - 2848725*x^9 + 25958664*x^8 - 124445806*x^7 + 333271318*x^6 - 536215885*x^5 + 540397440*x^4 - 344500525*x^3 + 135184650*x^2 - 29851500*x + 2843000, 1)
 

Normalized defining polynomial

\( x^{15} + 36 x^{13} - 144 x^{12} - 32503 x^{11} + 261104 x^{10} - 2848725 x^{9} + 25958664 x^{8} - 124445806 x^{7} + 333271318 x^{6} - 536215885 x^{5} + 540397440 x^{4} - 344500525 x^{3} + 135184650 x^{2} - 29851500 x + 2843000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1556733003528499930254458732658332584781471296000000=2^{12}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 71^{2}\cdot 337^{2}\cdot 2843^{4}\cdot 116719^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2587.11$
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:  $2, 5, 7, 13, 71, 337, 2843, 116719$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{1}{25} a^{4} + \frac{8}{25} a^{3} + \frac{4}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} - \frac{1}{25} a^{6} - \frac{1}{25} a^{5} + \frac{8}{25} a^{4} + \frac{4}{25} a^{3} - \frac{2}{5} a$, $\frac{1}{325} a^{10} - \frac{1}{325} a^{8} + \frac{4}{325} a^{7} - \frac{16}{325} a^{6} - \frac{32}{325} a^{5} + \frac{159}{325} a^{4} + \frac{9}{65} a^{3} + \frac{18}{65} a^{2} - \frac{3}{13} a + \frac{6}{13}$, $\frac{1}{1625} a^{11} - \frac{14}{1625} a^{9} - \frac{9}{1625} a^{8} + \frac{62}{1625} a^{7} - \frac{71}{1625} a^{6} - \frac{3}{65} a^{5} - \frac{371}{1625} a^{4} + \frac{324}{1625} a^{3} + \frac{718}{1625} a^{2} - \frac{74}{325} a - \frac{3}{25}$, $\frac{1}{8125} a^{12} - \frac{9}{8125} a^{10} + \frac{56}{8125} a^{9} - \frac{73}{8125} a^{8} + \frac{534}{8125} a^{7} - \frac{83}{1625} a^{6} + \frac{184}{8125} a^{5} - \frac{2456}{8125} a^{4} - \frac{3412}{8125} a^{3} - \frac{673}{1625} a^{2} + \frac{406}{1625} a + \frac{147}{325}$, $\frac{1}{32500000} a^{13} + \frac{31}{1625000} a^{12} - \frac{439}{2031250} a^{11} + \frac{1243}{1015625} a^{10} + \frac{543657}{32500000} a^{9} - \frac{46129}{8125000} a^{8} + \frac{276427}{6500000} a^{7} - \frac{796069}{8125000} a^{6} - \frac{32013}{16250000} a^{5} + \frac{7129079}{16250000} a^{4} - \frac{1987993}{6500000} a^{3} - \frac{410577}{1625000} a^{2} - \frac{418257}{1300000} a + \frac{312991}{650000}$, $\frac{1}{1040000000} a^{14} - \frac{1}{104000000} a^{13} + \frac{1297}{130000000} a^{12} - \frac{4847}{32500000} a^{11} - \frac{1287223}{1040000000} a^{10} + \frac{10219787}{520000000} a^{9} - \frac{295357}{208000000} a^{8} - \frac{44998663}{520000000} a^{7} - \frac{47385073}{520000000} a^{6} + \frac{17633269}{520000000} a^{5} - \frac{8741501}{208000000} a^{4} + \frac{19189041}{104000000} a^{3} - \frac{18150649}{41600000} a^{2} - \frac{3451627}{10400000} a + \frac{107167}{2080000}$

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:  $12$
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:  \( 263754998565000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T98:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 71 conjugacy class representatives for [1/2.S(5)^3]3 are not computed
Character table for [1/2.S(5)^3]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R $15$ R $15$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.10$x^{12} - 6 x^{10} + 23 x^{8} - 28 x^{6} - 9 x^{4} - 30 x^{2} - 15$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
71Data not computed
337Data not computed
2843Data not computed
116719Data not computed