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Properties

Label 16.0.10455582754...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{12}\cdot 79^{4}$
Root discriminant $56.39$
Ramified primes $2, 5, 79$
Class number $2720$ (GRH)
Class group $[2, 2, 2, 2, 170]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1069756, 899376, 1960928, 35312, 1448468, -743520, 874144, -402184, 224498, -69432, 25196, -5224, 1478, -204, 52, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 52*x^14 - 204*x^13 + 1478*x^12 - 5224*x^11 + 25196*x^10 - 69432*x^9 + 224498*x^8 - 402184*x^7 + 874144*x^6 - 743520*x^5 + 1448468*x^4 + 35312*x^3 + 1960928*x^2 + 899376*x + 1069756)
 
gp: K = bnfinit(x^16 - 4*x^15 + 52*x^14 - 204*x^13 + 1478*x^12 - 5224*x^11 + 25196*x^10 - 69432*x^9 + 224498*x^8 - 402184*x^7 + 874144*x^6 - 743520*x^5 + 1448468*x^4 + 35312*x^3 + 1960928*x^2 + 899376*x + 1069756, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 52 x^{14} - 204 x^{13} + 1478 x^{12} - 5224 x^{11} + 25196 x^{10} - 69432 x^{9} + 224498 x^{8} - 402184 x^{7} + 874144 x^{6} - 743520 x^{5} + 1448468 x^{4} + 35312 x^{3} + 1960928 x^{2} + 899376 x + 1069756 \)

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:  \(10455582754471936000000000000=2^{40}\cdot 5^{12}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.39$
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:  $2, 5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{5058} a^{14} + \frac{41}{843} a^{13} - \frac{69}{281} a^{12} + \frac{19}{562} a^{11} - \frac{224}{2529} a^{10} - \frac{265}{2529} a^{9} + \frac{559}{5058} a^{8} - \frac{73}{281} a^{7} - \frac{1084}{2529} a^{6} + \frac{304}{843} a^{5} + \frac{202}{843} a^{4} + \frac{280}{843} a^{3} - \frac{797}{2529} a^{2} + \frac{887}{2529} a - \frac{1153}{2529}$, $\frac{1}{179743387813414842435675821215284777697494} a^{15} + \frac{1550125351635243067166615773551508471}{179743387813414842435675821215284777697494} a^{14} - \frac{4390492974941732128520647744509177914717}{59914462604471614145225273738428259232498} a^{13} - \frac{849348046444720400292748795983719777266}{9985743767411935690870878956404709872083} a^{12} + \frac{11228457119573322646114410055012856061521}{179743387813414842435675821215284777697494} a^{11} - \frac{97549910490617134311465604438481369}{14760892486935603386357544651004744822} a^{10} - \frac{7619510087125473444961659309228419731955}{89871693906707421217837910607642388848747} a^{9} - \frac{21857216797526439771413162336902024878737}{179743387813414842435675821215284777697494} a^{8} - \frac{22455194682217206425399140740798679367611}{89871693906707421217837910607642388848747} a^{7} + \frac{36530811316363202784706510573947621787676}{89871693906707421217837910607642388848747} a^{6} + \frac{2872093057721628356531537886117456935723}{29957231302235807072612636869214129616249} a^{5} - \frac{1159314026465263973926977810675556800401}{2723384663839618824782966988110375419659} a^{4} + \frac{10387362778028481113964480751802160412957}{89871693906707421217837910607642388848747} a^{3} - \frac{143786275755798529987816324496001003991}{730664178103312367624698460224734868689} a^{2} - \frac{12647034605705849997944192514856093182443}{89871693906707421217837910607642388848747} a - \frac{10255114368611774502108311362358915782717}{89871693906707421217837910607642388848747}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{170}$, which has order $2720$ (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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$