# Properties

 Label 15.5.154972454814106259.1 Degree $15$ Signature $[5, 5]$ Discriminant $-\,11^{13}\cdot 67^{2}$ Root discriminant $14.00$ Ramified primes $11, 67$ Class number $1$ Class group Trivial Galois group 15T44

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -7, 27, -7, -34, 28, 8, -15, 2, -1, 1, 1, -2, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1)

gp: K = bnfinit(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1, 1)

## Normalizeddefining polynomial

$$x^{15} - x^{14} - 2 x^{13} + x^{12} + x^{11} - x^{10} + 2 x^{9} - 15 x^{8} + 8 x^{7} + 28 x^{6} - 34 x^{5} - 7 x^{4} + 27 x^{3} - 7 x^{2} - 3 x + 1$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $15$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[5, 5]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$-154972454814106259=-\,11^{13}\cdot 67^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $14.00$ 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: $11, 67$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1375967} a^{14} + \frac{175735}{1375967} a^{13} - \frac{613357}{1375967} a^{12} + \frac{221128}{1375967} a^{11} + \frac{90195}{1375967} a^{10} - \frac{631321}{1375967} a^{9} - \frac{232077}{1375967} a^{8} - \frac{621807}{1375967} a^{7} - \frac{79672}{1375967} a^{6} + \frac{601628}{1375967} a^{5} - \frac{230139}{1375967} a^{4} + \frac{90720}{1375967} a^{3} - \frac{559682}{1375967} a^{2} + \frac{597135}{1375967} a - \frac{6898}{1375967}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $9$ 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: $$492.79424719$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 2430 The 39 conjugacy class representatives for [3^5:2]5 Character table for [3^5:2]5 is not computed

## Intermediate fields

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

## Sibling fields

 Degree 15 siblings: data not computed Degree 30 siblings: data not computed Degree 45 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ $15$ $15$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5} 11.10.9.7x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
67Data not computed