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Properties

Label 16.0.26968313608...625.27
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8629655, -1890290, 6353305, -2418440, 2870551, -927958, 673820, -166580, 93021, -15125, 7911, -790, 536, -40, 30, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 30*x^14 - 40*x^13 + 536*x^12 - 790*x^11 + 7911*x^10 - 15125*x^9 + 93021*x^8 - 166580*x^7 + 673820*x^6 - 927958*x^5 + 2870551*x^4 - 2418440*x^3 + 6353305*x^2 - 1890290*x + 8629655)
 
gp: K = bnfinit(x^16 - 2*x^15 + 30*x^14 - 40*x^13 + 536*x^12 - 790*x^11 + 7911*x^10 - 15125*x^9 + 93021*x^8 - 166580*x^7 + 673820*x^6 - 927958*x^5 + 2870551*x^4 - 2418440*x^3 + 6353305*x^2 - 1890290*x + 8629655, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 30 x^{14} - 40 x^{13} + 536 x^{12} - 790 x^{11} + 7911 x^{10} - 15125 x^{9} + 93021 x^{8} - 166580 x^{7} + 673820 x^{6} - 927958 x^{5} + 2870551 x^{4} - 2418440 x^{3} + 6353305 x^{2} - 1890290 x + 8629655 \)

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:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{89269480961220101924103458633348932857121} a^{15} - \frac{10993799679670801296823561004759371926666}{89269480961220101924103458633348932857121} a^{14} - \frac{5234160493863812482113590581262855528080}{89269480961220101924103458633348932857121} a^{13} - \frac{11446413344057237152487305799070003649251}{89269480961220101924103458633348932857121} a^{12} - \frac{28422657235040464148748356294413935556128}{89269480961220101924103458633348932857121} a^{11} + \frac{32034800163315633859264397847309873337685}{89269480961220101924103458633348932857121} a^{10} - \frac{37417056214501850411834561909389391573629}{89269480961220101924103458633348932857121} a^{9} + \frac{10882244297262630210422291373471612263498}{89269480961220101924103458633348932857121} a^{8} - \frac{27690530905980085178850306333158818786543}{89269480961220101924103458633348932857121} a^{7} + \frac{2655186250471568067683920606056022661358}{89269480961220101924103458633348932857121} a^{6} - \frac{1651314128059775963721069976136804552355}{89269480961220101924103458633348932857121} a^{5} + \frac{34385335447043401040943774151813490359087}{89269480961220101924103458633348932857121} a^{4} - \frac{16407865256766797830745163948753355477941}{89269480961220101924103458633348932857121} a^{3} + \frac{15002637820291811110619238310224447532638}{89269480961220101924103458633348932857121} a^{2} - \frac{18516399247204524844724375363930661543172}{89269480961220101924103458633348932857121} a - \frac{14241369810873012689486167246123871888572}{89269480961220101924103458633348932857121}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 1166281.37271 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 4.0.1275125.2, 4.0.12625.1, 8.4.643938125.1, 8.4.164220320328125.2, 8.0.1625943765625.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed