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Properties

Label 16.0.18357500856...5264.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 113^{4}$
Root discriminant $28.44$
Ramified primes $2, 113$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![953, 1624, 3224, 4240, 6060, 5680, 3328, 1112, 144, 72, 56, -16, -20, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^13 - 20*x^12 - 16*x^11 + 56*x^10 + 72*x^9 + 144*x^8 + 1112*x^7 + 3328*x^6 + 5680*x^5 + 6060*x^4 + 4240*x^3 + 3224*x^2 + 1624*x + 953)
 
gp: K = bnfinit(x^16 - 8*x^13 - 20*x^12 - 16*x^11 + 56*x^10 + 72*x^9 + 144*x^8 + 1112*x^7 + 3328*x^6 + 5680*x^5 + 6060*x^4 + 4240*x^3 + 3224*x^2 + 1624*x + 953, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{13} - 20 x^{12} - 16 x^{11} + 56 x^{10} + 72 x^{9} + 144 x^{8} + 1112 x^{7} + 3328 x^{6} + 5680 x^{5} + 6060 x^{4} + 4240 x^{3} + 3224 x^{2} + 1624 x + 953 \)

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:  \(183575008560835685515264=2^{50}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.44$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{7}{23} a^{13} + \frac{11}{23} a^{12} - \frac{3}{23} a^{11} - \frac{3}{23} a^{10} + \frac{4}{23} a^{9} + \frac{4}{23} a^{8} + \frac{7}{23} a^{7} - \frac{11}{23} a^{6} + \frac{3}{23} a^{5} - \frac{1}{23} a^{4} + \frac{7}{23} a^{3} - \frac{5}{23} a - \frac{7}{23}$, $\frac{1}{938984471666607455485398335} a^{15} - \frac{8653314063594987605187179}{938984471666607455485398335} a^{14} - \frac{220673015522036749401000294}{938984471666607455485398335} a^{13} - \frac{84890392216659513266848972}{938984471666607455485398335} a^{12} + \frac{299113825155254195363620163}{938984471666607455485398335} a^{11} - \frac{280113584413783600106428428}{938984471666607455485398335} a^{10} - \frac{6471781551345554506744937}{938984471666607455485398335} a^{9} + \frac{3571790657786820831196700}{8165082362318325699873029} a^{8} - \frac{446004370254240720906294396}{938984471666607455485398335} a^{7} - \frac{455348588380488517862045394}{938984471666607455485398335} a^{6} - \frac{122177360900244133167439221}{938984471666607455485398335} a^{5} + \frac{103398161935208163774440834}{938984471666607455485398335} a^{4} - \frac{144813108012445457332817601}{938984471666607455485398335} a^{3} - \frac{34953789841211048375865121}{938984471666607455485398335} a^{2} - \frac{425307539935592876428288782}{938984471666607455485398335} a - \frac{203959845476778400244313798}{938984471666607455485398335}$

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:  $7$
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:  \( 90719.9112667 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1024.1, 8.4.473956352.4

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
113Data not computed