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Properties

Label 15.15.7447775355...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{16}\cdot 5^{15}\cdot 7^{12}$
Root discriminant $133.29$
Ramified primes $2, 3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T41

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-210244, 1700980, -4274080, 2177500, 4119760, -2594116, -1750400, 699700, 307730, -81630, -22891, 4795, 680, -130, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 130*x^13 + 680*x^12 + 4795*x^11 - 22891*x^10 - 81630*x^9 + 307730*x^8 + 699700*x^7 - 1750400*x^6 - 2594116*x^5 + 4119760*x^4 + 2177500*x^3 - 4274080*x^2 + 1700980*x - 210244)
 
gp: K = bnfinit(x^15 - 5*x^14 - 130*x^13 + 680*x^12 + 4795*x^11 - 22891*x^10 - 81630*x^9 + 307730*x^8 + 699700*x^7 - 1750400*x^6 - 2594116*x^5 + 4119760*x^4 + 2177500*x^3 - 4274080*x^2 + 1700980*x - 210244, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 130 x^{13} + 680 x^{12} + 4795 x^{11} - 22891 x^{10} - 81630 x^{9} + 307730 x^{8} + 699700 x^{7} - 1750400 x^{6} - 2594116 x^{5} + 4119760 x^{4} + 2177500 x^{3} - 4274080 x^{2} + 1700980 x - 210244 \)

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:  \(74477753552789740125000000000000=2^{12}\cdot 3^{16}\cdot 5^{15}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.29$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{2}{5} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} + \frac{1}{10} a^{7} - \frac{1}{5} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{11} - \frac{1}{10} a^{9} - \frac{1}{20} a^{8} - \frac{1}{20} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{10} a^{8} + \frac{1}{4} a^{7} - \frac{2}{5} a^{6} + \frac{3}{10} a^{5} + \frac{3}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{611320401926445433979337963687688820} a^{14} - \frac{13862922040952699881875405503499207}{611320401926445433979337963687688820} a^{13} + \frac{14987857729210411378404257095993369}{611320401926445433979337963687688820} a^{12} - \frac{21705338653458117750378261716182651}{611320401926445433979337963687688820} a^{11} - \frac{14757032094771977533216709089334253}{611320401926445433979337963687688820} a^{10} - \frac{10459297752015052943237528848521051}{611320401926445433979337963687688820} a^{9} - \frac{25325022496746467303896199874677931}{611320401926445433979337963687688820} a^{8} + \frac{66865346157132972688702981668050983}{611320401926445433979337963687688820} a^{7} - \frac{23091431123343471991424828216681739}{61132040192644543397933796368768882} a^{6} - \frac{33509771884599476938759287411808787}{152830100481611358494834490921922205} a^{5} - \frac{21403498915840951216522987521072471}{61132040192644543397933796368768882} a^{4} + \frac{5114040572612070173980442363163809}{305660200963222716989668981843844410} a^{3} + \frac{21580067429010824097533246873279909}{305660200963222716989668981843844410} a^{2} - \frac{24395922770282900609928626749590241}{152830100481611358494834490921922205} a + \frac{15121958902106858645002282114067919}{152830100481611358494834490921922205}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 477769896295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T41:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1620
The 21 conjugacy class representatives for [3^4]F(5)
Character table for [3^4]F(5) is not computed

Intermediate fields

5.5.2450000.1

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 30 siblings: data not computed
Degree 45 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 R R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.16.9$x^{12} + 117 x^{11} + 81 x^{10} - 39 x^{9} + 18 x^{8} - 108 x^{7} + 63 x^{6} - 54 x^{5} - 81 x^{4} - 54 x^{3} - 81 x^{2} - 81$$3$$4$$16$12T73$[2, 2]^{12}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.10.4$x^{12} - 7 x^{6} + 147$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$