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Properties

Label 15.5.69696371772...0743.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,23^{5}\cdot 101^{8}$
Root discriminant $33.33$
Ramified primes $23, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 16, -30, -91, 1067, -290, -920, 1108, 60, -737, 528, -108, -43, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 21*x^13 - 43*x^12 - 108*x^11 + 528*x^10 - 737*x^9 + 60*x^8 + 1108*x^7 - 920*x^6 - 290*x^5 + 1067*x^4 - 91*x^3 - 30*x^2 + 16*x - 8)
 
gp: K = bnfinit(x^15 - 5*x^14 + 21*x^13 - 43*x^12 - 108*x^11 + 528*x^10 - 737*x^9 + 60*x^8 + 1108*x^7 - 920*x^6 - 290*x^5 + 1067*x^4 - 91*x^3 - 30*x^2 + 16*x - 8, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 21 x^{13} - 43 x^{12} - 108 x^{11} + 528 x^{10} - 737 x^{9} + 60 x^{8} + 1108 x^{7} - 920 x^{6} - 290 x^{5} + 1067 x^{4} - 91 x^{3} - 30 x^{2} + 16 x - 8 \)

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:  \(-69696371772723539550743=-\,23^{5}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.33$
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:  $23, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{36} a^{11} - \frac{1}{12} a^{10} + \frac{1}{36} a^{8} + \frac{1}{36} a^{7} - \frac{1}{36} a^{6} - \frac{11}{36} a^{5} - \frac{1}{3} a^{4} - \frac{1}{18} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{4}{9}$, $\frac{1}{108} a^{12} + \frac{1}{108} a^{11} + \frac{1}{18} a^{10} + \frac{7}{108} a^{9} + \frac{5}{108} a^{8} - \frac{1}{12} a^{7} - \frac{13}{36} a^{6} - \frac{2}{27} a^{5} - \frac{1}{54} a^{4} + \frac{49}{108} a^{3} + \frac{10}{27} a + \frac{7}{27}$, $\frac{1}{324} a^{13} - \frac{1}{324} a^{12} + \frac{1}{81} a^{11} + \frac{13}{324} a^{10} - \frac{1}{12} a^{9} - \frac{19}{324} a^{8} - \frac{1}{108} a^{7} + \frac{40}{81} a^{6} + \frac{8}{81} a^{5} - \frac{145}{324} a^{4} - \frac{49}{162} a^{3} - \frac{8}{81} a^{2} + \frac{1}{162} a + \frac{4}{81}$, $\frac{1}{17312101053516} a^{14} - \frac{90670375}{106864821318} a^{13} + \frac{20817540511}{5770700351172} a^{12} + \frac{32158284097}{8656050526758} a^{11} - \frac{1187713926293}{17312101053516} a^{10} + \frac{274995475382}{4328025263379} a^{9} + \frac{1284504597761}{17312101053516} a^{8} - \frac{527374860361}{8656050526758} a^{7} + \frac{2182033559647}{5770700351172} a^{6} + \frac{2343995351651}{8656050526758} a^{5} - \frac{500087020063}{1923566783724} a^{4} - \frac{2021930771815}{4328025263379} a^{3} + \frac{2046500336465}{5770700351172} a^{2} + \frac{121946164349}{961783391862} a + \frac{1087543986646}{4328025263379}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 577959.629224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

3.1.23.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.5.4.3$x^{5} - 404$$5$$1$$4$$C_5$$[\ ]_{5}$
101.5.4.3$x^{5} - 404$$5$$1$$4$$C_5$$[\ ]_{5}$