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Properties

Label 15.15.1103798189...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{5}\cdot 61^{3}\cdot 109^{2}\cdot 397^{3}\cdot 1429759^{2}$
Root discriminant $252.85$
Ramified primes $2, 5, 61, 109, 397, 1429759$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T83

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65199753925, -65562916998, -1626100824, 16802789919, -1625667330, -1822041101, 181291735, 107371802, -6928850, -3483491, 82577, 55169, -84, -393, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 393*x^13 - 84*x^12 + 55169*x^11 + 82577*x^10 - 3483491*x^9 - 6928850*x^8 + 107371802*x^7 + 181291735*x^6 - 1822041101*x^5 - 1625667330*x^4 + 16802789919*x^3 - 1626100824*x^2 - 65562916998*x + 65199753925)
 
gp: K = bnfinit(x^15 - 2*x^14 - 393*x^13 - 84*x^12 + 55169*x^11 + 82577*x^10 - 3483491*x^9 - 6928850*x^8 + 107371802*x^7 + 181291735*x^6 - 1822041101*x^5 - 1625667330*x^4 + 16802789919*x^3 - 1626100824*x^2 - 65562916998*x + 65199753925, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 393 x^{13} - 84 x^{12} + 55169 x^{11} + 82577 x^{10} - 3483491 x^{9} - 6928850 x^{8} + 107371802 x^{7} + 181291735 x^{6} - 1822041101 x^{5} - 1625667330 x^{4} + 16802789919 x^{3} - 1626100824 x^{2} - 65562916998 x + 65199753925 \)

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:  \(1103798189912877152281465916777600000=2^{10}\cdot 5^{5}\cdot 61^{3}\cdot 109^{2}\cdot 397^{3}\cdot 1429759^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.85$
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:  $2, 5, 61, 109, 397, 1429759$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{1949919577317780171978070126745053726781769616557856812705458} a^{14} - \frac{8510630221943906973947695060255977626084705001185153554939}{1949919577317780171978070126745053726781769616557856812705458} a^{13} - \frac{465667924022167719011019341285022780412735146296148187355389}{1949919577317780171978070126745053726781769616557856812705458} a^{12} - \frac{259344690203783886396592631836126802307134145712192214726047}{1949919577317780171978070126745053726781769616557856812705458} a^{11} + \frac{410841772781025904852843348044553663810228395302260631347059}{1949919577317780171978070126745053726781769616557856812705458} a^{10} + \frac{199024306958688088120461307441457141305657033998872328751381}{1949919577317780171978070126745053726781769616557856812705458} a^{9} - \frac{282793031424979542826611541832100378816655796627535075434857}{1949919577317780171978070126745053726781769616557856812705458} a^{8} - \frac{171154632443886836439143863862361256605463069335127935705877}{1949919577317780171978070126745053726781769616557856812705458} a^{7} - \frac{121073401133732933722371351156075750566845480488889366381584}{974959788658890085989035063372526863390884808278928406352729} a^{6} - \frac{446439799565470971037318193844506481321470971803711004541113}{974959788658890085989035063372526863390884808278928406352729} a^{5} + \frac{252015877837035351094570217264120548746933481631677301349005}{1949919577317780171978070126745053726781769616557856812705458} a^{4} + \frac{122463815074269005820219797974150067417500587640945179694265}{1949919577317780171978070126745053726781769616557856812705458} a^{3} + \frac{255113887647665425471809313001065182549398120543174442989215}{1949919577317780171978070126745053726781769616557856812705458} a^{2} - \frac{253228001280017214301573810478648626669275652512390315603081}{1949919577317780171978070126745053726781769616557856812705458} a + \frac{348628541660135915835674578731620982780797823327546386951332}{974959788658890085989035063372526863390884808278928406352729}$

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:  \( 7320058498710 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T83:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 58320
The 72 conjugacy class representatives for [3^5:2]S(5) are not computed
Character table for [3^5:2]S(5) is not computed

Intermediate fields

5.5.24217.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed
$109$109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
397Data not computed
1429759Data not computed