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Properties

Label 16.0.87997506881...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 547889^{4}$
Root discriminant $74.39$
Ramified primes $5, 547889$
Class number $15795$ (GRH)
Class group $[15795]$ (GRH)
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61523369, -24641603, 37525970, -25438633, 14806730, -8562175, 4288394, -1612435, 717275, -183922, 67230, -12135, 3516, -420, 95, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 95*x^14 - 420*x^13 + 3516*x^12 - 12135*x^11 + 67230*x^10 - 183922*x^9 + 717275*x^8 - 1612435*x^7 + 4288394*x^6 - 8562175*x^5 + 14806730*x^4 - 25438633*x^3 + 37525970*x^2 - 24641603*x + 61523369)
 
gp: K = bnfinit(x^16 - 6*x^15 + 95*x^14 - 420*x^13 + 3516*x^12 - 12135*x^11 + 67230*x^10 - 183922*x^9 + 717275*x^8 - 1612435*x^7 + 4288394*x^6 - 8562175*x^5 + 14806730*x^4 - 25438633*x^3 + 37525970*x^2 - 24641603*x + 61523369, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 95 x^{14} - 420 x^{13} + 3516 x^{12} - 12135 x^{11} + 67230 x^{10} - 183922 x^{9} + 717275 x^{8} - 1612435 x^{7} + 4288394 x^{6} - 8562175 x^{5} + 14806730 x^{4} - 25438633 x^{3} + 37525970 x^{2} - 24641603 x + 61523369 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(879975068812771568896884765625=5^{10}\cdot 547889^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.39$
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:  $5, 547889$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{275} a^{14} - \frac{102}{275} a^{13} + \frac{3}{275} a^{12} + \frac{27}{55} a^{11} - \frac{46}{275} a^{10} - \frac{59}{275} a^{9} + \frac{108}{275} a^{8} + \frac{16}{275} a^{7} - \frac{8}{275} a^{6} + \frac{89}{275} a^{5} - \frac{103}{275} a^{4} - \frac{13}{275} a^{3} + \frac{1}{5} a^{2} - \frac{21}{275} a - \frac{109}{275}$, $\frac{1}{70028822920394226117983113053692315003788197973525} a^{15} + \frac{77165132221899540181588792788771812945808960849}{70028822920394226117983113053692315003788197973525} a^{14} + \frac{10811531118085656535460952135501116887974070035951}{70028822920394226117983113053692315003788197973525} a^{13} + \frac{11524194621113868191053548549616342086906480411813}{70028822920394226117983113053692315003788197973525} a^{12} - \frac{4451735661488590862727659071829595129598078888011}{70028822920394226117983113053692315003788197973525} a^{11} - \frac{2590452868653016228555616940770361555905661766911}{14005764584078845223596622610738463000757639594705} a^{10} + \frac{12345616908358246744310160465168342953213684774399}{70028822920394226117983113053692315003788197973525} a^{9} - \frac{21979475983345649304412412441258358147340441659951}{70028822920394226117983113053692315003788197973525} a^{8} + \frac{2033699207743381016304523620909425099514560643278}{6366256629126747828907555732153846818526199815775} a^{7} + \frac{3308164207252706230832680160214738397539065330206}{70028822920394226117983113053692315003788197973525} a^{6} - \frac{6292990097929804501036998064267772539575482932739}{70028822920394226117983113053692315003788197973525} a^{5} + \frac{30216556698644440847139580110989293038989593558009}{70028822920394226117983113053692315003788197973525} a^{4} + \frac{1115877574263032867915987849996944749902660814767}{70028822920394226117983113053692315003788197973525} a^{3} + \frac{22293260033025898532832851639315464979564263365809}{70028822920394226117983113053692315003788197973525} a^{2} + \frac{1838557302894950125517991293844936508278208067369}{14005764584078845223596622610738463000757639594705} a - \frac{10572634329237641713550123919012011237028312553659}{70028822920394226117983113053692315003788197973525}$

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

Class group and class number

$C_{15795}$, which has order $15795$ (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:  $7$
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:  \( 9516.84702281 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.13697225.1, 8.8.1712153125.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
547889Data not computed