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Properties

Label 15.15.4757687700...0625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 7^{10}\cdot 41^{4}$
Root discriminant $129.37$
Ramified primes $5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5^2:C_3$ (as 15T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![84419, 578100, 485235, -1204580, -1649225, 147955, 892410, 188340, -157290, -52050, 8819, 3950, -145, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 110*x^13 - 145*x^12 + 3950*x^11 + 8819*x^10 - 52050*x^9 - 157290*x^8 + 188340*x^7 + 892410*x^6 + 147955*x^5 - 1649225*x^4 - 1204580*x^3 + 485235*x^2 + 578100*x + 84419)
 
gp: K = bnfinit(x^15 - 110*x^13 - 145*x^12 + 3950*x^11 + 8819*x^10 - 52050*x^9 - 157290*x^8 + 188340*x^7 + 892410*x^6 + 147955*x^5 - 1649225*x^4 - 1204580*x^3 + 485235*x^2 + 578100*x + 84419, 1)
 

Normalized defining polynomial

\( x^{15} - 110 x^{13} - 145 x^{12} + 3950 x^{11} + 8819 x^{10} - 52050 x^{9} - 157290 x^{8} + 188340 x^{7} + 892410 x^{6} + 147955 x^{5} - 1649225 x^{4} - 1204580 x^{3} + 485235 x^{2} + 578100 x + 84419 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47576877003281652927398681640625=5^{24}\cdot 7^{10}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.37$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4057272284333820575700974128560177204853} a^{14} - \frac{1242613621176630978480101266460601441147}{4057272284333820575700974128560177204853} a^{13} + \frac{675310823649245699888080611118631431061}{4057272284333820575700974128560177204853} a^{12} + \frac{750609017113832675615695732867428697889}{4057272284333820575700974128560177204853} a^{11} - \frac{1841158514979336099914233195067334908876}{4057272284333820575700974128560177204853} a^{10} + \frac{1106677756102271966632680692460910919514}{4057272284333820575700974128560177204853} a^{9} + \frac{232153532620226623931817294962128124787}{4057272284333820575700974128560177204853} a^{8} + \frac{733192387236441552422525617027410034433}{4057272284333820575700974128560177204853} a^{7} + \frac{957078044493665146450250340731897691005}{4057272284333820575700974128560177204853} a^{6} - \frac{1343115051022933779620669967735439157894}{4057272284333820575700974128560177204853} a^{5} - \frac{528459354158084321563373254782646238110}{4057272284333820575700974128560177204853} a^{4} + \frac{65828693167952102168864650471241178636}{4057272284333820575700974128560177204853} a^{3} + \frac{311543970466433840298190508072501933526}{4057272284333820575700974128560177204853} a^{2} - \frac{1760484383254365699017511111685848478938}{4057272284333820575700974128560177204853} a + \frac{45498207654619527220055489871656342564}{4057272284333820575700974128560177204853}$

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:  $14$
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:  \( 94377547844.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5^2:C_3$ (as 15T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 375
The 55 conjugacy class representatives for $C_5\times C_5^2:C_3$ are not computed
Character table for $C_5\times C_5^2:C_3$ 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 15 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 $15$ $15$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ $15$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ $15$ $15$

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
5Data not computed
7Data not computed
41Data not computed