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Properties

Label 16.4.87678952778...7713.4
Degree $16$
Signature $[4, 6]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9255611219, 725765798, 956191714, 134079573, 250869331, 16338438, 9623711, 2235206, 1273077, -10925, 18570, 1891, -1290, 1, -45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 45*x^14 + x^13 - 1290*x^12 + 1891*x^11 + 18570*x^10 - 10925*x^9 + 1273077*x^8 + 2235206*x^7 + 9623711*x^6 + 16338438*x^5 + 250869331*x^4 + 134079573*x^3 + 956191714*x^2 + 725765798*x + 9255611219)
 
gp: K = bnfinit(x^16 - x^15 - 45*x^14 + x^13 - 1290*x^12 + 1891*x^11 + 18570*x^10 - 10925*x^9 + 1273077*x^8 + 2235206*x^7 + 9623711*x^6 + 16338438*x^5 + 250869331*x^4 + 134079573*x^3 + 956191714*x^2 + 725765798*x + 9255611219, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 45 x^{14} + x^{13} - 1290 x^{12} + 1891 x^{11} + 18570 x^{10} - 10925 x^{9} + 1273077 x^{8} + 2235206 x^{7} + 9623711 x^{6} + 16338438 x^{5} + 250869331 x^{4} + 134079573 x^{3} + 956191714 x^{2} + 725765798 x + 9255611219 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{244} a^{14} + \frac{5}{61} a^{13} - \frac{6}{61} a^{12} + \frac{57}{244} a^{11} - \frac{33}{244} a^{10} - \frac{73}{244} a^{9} + \frac{35}{122} a^{8} - \frac{23}{61} a^{7} + \frac{35}{244} a^{6} + \frac{33}{244} a^{5} + \frac{11}{244} a^{4} + \frac{18}{61} a^{3} + \frac{25}{122} a^{2} + \frac{3}{244} a - \frac{1}{244}$, $\frac{1}{1166781462244526975753632176969185886601184005540601486084486633828} a^{15} - \frac{378480431318889465842575113495577913461290065459482360599360079}{1166781462244526975753632176969185886601184005540601486084486633828} a^{14} + \frac{4189373281057994431010493182578116551735058553954352463072659298}{291695365561131743938408044242296471650296001385150371521121658457} a^{13} + \frac{35101477554845501208147699766624992438047955559028770932558842405}{1166781462244526975753632176969185886601184005540601486084486633828} a^{12} + \frac{28396381331987088437207403538090772017402660004536766234543804483}{291695365561131743938408044242296471650296001385150371521121658457} a^{11} + \frac{665871080233058165071549195833404740832975486164713415515396313}{291695365561131743938408044242296471650296001385150371521121658457} a^{10} - \frac{272173595665213439231098347198543263162300181485499926953067371327}{1166781462244526975753632176969185886601184005540601486084486633828} a^{9} - \frac{251872795890618072741976747297050534001051679012223305144551839103}{583390731122263487876816088484592943300592002770300743042243316914} a^{8} + \frac{1377349676305754058175730147784634281067560253555834031921615411}{11552291705391356195580516603655305807932514906342588971133531028} a^{7} - \frac{103170767809118539325974979719474360530270531421703698778384317091}{291695365561131743938408044242296471650296001385150371521121658457} a^{6} - \frac{178717969220964990192831752754815972456027463937722361804061559313}{583390731122263487876816088484592943300592002770300743042243316914} a^{5} - \frac{357675068503622378621303820448017047175817271380923108547931781869}{1166781462244526975753632176969185886601184005540601486084486633828} a^{4} - \frac{69191576356000704542012782521441348422138644079108416894123253711}{583390731122263487876816088484592943300592002770300743042243316914} a^{3} + \frac{4033098965945654953882445646002443318201004013449767431054797441}{11552291705391356195580516603655305807932514906342588971133531028} a^{2} - \frac{42007290532992461998041585855201306406664968224470476637070909967}{583390731122263487876816088484592943300592002770300743042243316914} a + \frac{238361630758751849445325266601334077838840143456672450588185899529}{1166781462244526975753632176969185886601184005540601486084486633828}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 153057741363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
61Data not computed
97Data not computed