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Properties

Label 16.0.50548861754...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{12}$
Root discriminant $82.98$
Ramified primes $2, 3, 5, 7$
Class number $15488$ (GRH)
Class group $[2, 2, 2, 2, 22, 44]$ (GRH)
Galois group $(C_2 \times Q_8):C_2$ (as 16T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13379913, -3082824, 12532680, 868392, 2514456, 1297464, 338520, 183684, 136663, 920, 19600, -364, 1624, -112, 76, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 76*x^14 - 112*x^13 + 1624*x^12 - 364*x^11 + 19600*x^10 + 920*x^9 + 136663*x^8 + 183684*x^7 + 338520*x^6 + 1297464*x^5 + 2514456*x^4 + 868392*x^3 + 12532680*x^2 - 3082824*x + 13379913)
 
gp: K = bnfinit(x^16 - 4*x^15 + 76*x^14 - 112*x^13 + 1624*x^12 - 364*x^11 + 19600*x^10 + 920*x^9 + 136663*x^8 + 183684*x^7 + 338520*x^6 + 1297464*x^5 + 2514456*x^4 + 868392*x^3 + 12532680*x^2 - 3082824*x + 13379913, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 76 x^{14} - 112 x^{13} + 1624 x^{12} - 364 x^{11} + 19600 x^{10} + 920 x^{9} + 136663 x^{8} + 183684 x^{7} + 338520 x^{6} + 1297464 x^{5} + 2514456 x^{4} + 868392 x^{3} + 12532680 x^{2} - 3082824 x + 13379913 \)

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:  \(5054886175427630513006837760000=2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.98$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{21} a^{8} - \frac{2}{21} a^{7} - \frac{2}{7} a^{6} - \frac{5}{21} a^{5} + \frac{1}{3} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{21} a^{9} - \frac{10}{21} a^{7} + \frac{4}{21} a^{6} - \frac{1}{7} a^{5} + \frac{5}{21} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{21} a^{10} + \frac{5}{21} a^{7} - \frac{1}{7} a^{5} + \frac{4}{21} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{21} a^{11} + \frac{10}{21} a^{7} + \frac{2}{7} a^{6} + \frac{8}{21} a^{5} - \frac{2}{21} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7}$, $\frac{1}{21} a^{12} + \frac{5}{21} a^{7} + \frac{5}{21} a^{6} + \frac{2}{7} a^{5} - \frac{10}{21} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{21} a^{13} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{5}{21} a^{4} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{987} a^{14} + \frac{1}{141} a^{13} + \frac{4}{329} a^{12} + \frac{20}{987} a^{11} + \frac{11}{987} a^{10} - \frac{8}{987} a^{9} + \frac{1}{141} a^{8} - \frac{75}{329} a^{7} + \frac{379}{987} a^{6} - \frac{96}{329} a^{5} + \frac{383}{987} a^{4} + \frac{145}{329} a^{3} + \frac{92}{329} a^{2} + \frac{9}{47} a - \frac{1}{7}$, $\frac{1}{276493566441021279381096765673824254251979691} a^{15} - \frac{94498519291190453697391391354562376981312}{276493566441021279381096765673824254251979691} a^{14} - \frac{484011662469906182063913045277328110411931}{92164522147007093127032255224608084750659897} a^{13} + \frac{2306027839002727350505993626058189869149864}{276493566441021279381096765673824254251979691} a^{12} - \frac{126590386052102056343769625641827049974969}{13166360306715299018147465032086869250094271} a^{11} - \frac{2596526188332456800975433306060691987319759}{276493566441021279381096765673824254251979691} a^{10} + \frac{137396321061985369258504421026758444261475}{276493566441021279381096765673824254251979691} a^{9} + \frac{368580723491211110182707982248596568591172}{39499080920145897054442395096260607750282813} a^{8} + \frac{24218403762332260777864674452999154888794810}{92164522147007093127032255224608084750659897} a^{7} - \frac{28341079969420923915306280740455080945486423}{92164522147007093127032255224608084750659897} a^{6} + \frac{16238880512696975232327118070657452670649154}{92164522147007093127032255224608084750659897} a^{5} + \frac{71480618566516074144588715187869334179757222}{276493566441021279381096765673824254251979691} a^{4} - \frac{6054630000555733653033766419032999107908696}{92164522147007093127032255224608084750659897} a^{3} - \frac{35804973075271458020164305758039543124568440}{92164522147007093127032255224608084750659897} a^{2} + \frac{2383227455656242185277599992902744512407341}{13166360306715299018147465032086869250094271} a + \frac{1148260015907664754207412744370078391095}{1960947279723555172915579898395916696822551}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{44}$, which has order $15488$ (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:  \( 32719.5368105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^3$ (as 16T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $(C_2 \times Q_8):C_2$
Character table for $(C_2 \times Q_8):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.12745506816.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$