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Properties

Label 16.8.87678952778...7713.4
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85935077, -9553740, -125389252, 50586939, -45938641, -42398902, 12299693, -3895029, 2148726, -78689, 50397, -7768, -1597, 507, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 68*x^14 + 507*x^13 - 1597*x^12 - 7768*x^11 + 50397*x^10 - 78689*x^9 + 2148726*x^8 - 3895029*x^7 + 12299693*x^6 - 42398902*x^5 - 45938641*x^4 + 50586939*x^3 - 125389252*x^2 - 9553740*x + 85935077)
 
gp: K = bnfinit(x^16 - 6*x^15 - 68*x^14 + 507*x^13 - 1597*x^12 - 7768*x^11 + 50397*x^10 - 78689*x^9 + 2148726*x^8 - 3895029*x^7 + 12299693*x^6 - 42398902*x^5 - 45938641*x^4 + 50586939*x^3 - 125389252*x^2 - 9553740*x + 85935077, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 68 x^{14} + 507 x^{13} - 1597 x^{12} - 7768 x^{11} + 50397 x^{10} - 78689 x^{9} + 2148726 x^{8} - 3895029 x^{7} + 12299693 x^{6} - 42398902 x^{5} - 45938641 x^{4} + 50586939 x^{3} - 125389252 x^{2} - 9553740 x + 85935077 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{412} a^{14} + \frac{12}{103} a^{13} + \frac{147}{412} a^{12} - \frac{179}{412} a^{11} - \frac{91}{206} a^{10} + \frac{7}{412} a^{9} + \frac{113}{412} a^{8} + \frac{89}{206} a^{7} - \frac{107}{412} a^{6} + \frac{27}{412} a^{5} + \frac{101}{206} a^{4} - \frac{105}{412} a^{3} + \frac{107}{412} a^{2} - \frac{73}{206} a - \frac{127}{412}$, $\frac{1}{3045175618493769082740406376006441417963924055720153005989946735896} a^{15} + \frac{759783402063660973847844113268812681111302334326244718415344467}{3045175618493769082740406376006441417963924055720153005989946735896} a^{14} + \frac{610809783974826403610868088359782626835021405173509644552602888307}{3045175618493769082740406376006441417963924055720153005989946735896} a^{13} + \frac{134074070227176390799473121481609254678242383310175602060842656455}{1522587809246884541370203188003220708981962027860076502994973367948} a^{12} - \frac{1354669547525366742441018980493547240794718378341990393400851344387}{3045175618493769082740406376006441417963924055720153005989946735896} a^{11} - \frac{462105216521994948308253943252887005717012355566650970181176205655}{3045175618493769082740406376006441417963924055720153005989946735896} a^{10} - \frac{265575338860164848870580604609828585143189154882057107547182158929}{1522587809246884541370203188003220708981962027860076502994973367948} a^{9} + \frac{819259520066632709791198069093049395424383819454495528952549269817}{3045175618493769082740406376006441417963924055720153005989946735896} a^{8} + \frac{1132859425542296880727289045611881791650545992080096751064949394307}{3045175618493769082740406376006441417963924055720153005989946735896} a^{7} + \frac{681440614680464779942375560938178877497096058349718926984646238627}{1522587809246884541370203188003220708981962027860076502994973367948} a^{6} + \frac{550819527873927888454404583365142471491044706529083754881757232223}{3045175618493769082740406376006441417963924055720153005989946735896} a^{5} - \frac{548557440804778330563545412600626116639041495453450140789459078507}{3045175618493769082740406376006441417963924055720153005989946735896} a^{4} - \frac{62220667039362989676358858488977741236032238359148409479282715535}{761293904623442270685101594001610354490981013930038251497486683974} a^{3} - \frac{1426743218381396402119469848806839974561443521643724525789449111765}{3045175618493769082740406376006441417963924055720153005989946735896} a^{2} + \frac{523450399950194107737731372350632764414093094941910730981328190451}{3045175618493769082740406376006441417963924055720153005989946735896} a - \frac{19377229741306272327841314996227294199065972423467272760467823513}{3045175618493769082740406376006441417963924055720153005989946735896}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 1486239938610 \) (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
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
97Data not computed