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Properties

Label 16.4.26703352420...3952.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{54}\cdot 7^{8}\cdot 137^{3}$
Root discriminant $69.05$
Ramified primes $2, 7, 137$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1461

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1244322, 1500912, 260704, 45600, 147356, 27728, 43160, 15928, -11682, 320, 864, -544, 440, -120, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 120*x^13 + 440*x^12 - 544*x^11 + 864*x^10 + 320*x^9 - 11682*x^8 + 15928*x^7 + 43160*x^6 + 27728*x^5 + 147356*x^4 + 45600*x^3 + 260704*x^2 + 1500912*x + 1244322)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 120*x^13 + 440*x^12 - 544*x^11 + 864*x^10 + 320*x^9 - 11682*x^8 + 15928*x^7 + 43160*x^6 + 27728*x^5 + 147356*x^4 + 45600*x^3 + 260704*x^2 + 1500912*x + 1244322, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 120 x^{13} + 440 x^{12} - 544 x^{11} + 864 x^{10} + 320 x^{9} - 11682 x^{8} + 15928 x^{7} + 43160 x^{6} + 27728 x^{5} + 147356 x^{4} + 45600 x^{3} + 260704 x^{2} + 1500912 x + 1244322 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(267033524201279981591516413952=2^{54}\cdot 7^{8}\cdot 137^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.05$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{1362760873221484918289678062664015532988759989} a^{15} - \frac{51098354203584086694709631124263998032855347}{1362760873221484918289678062664015532988759989} a^{14} - \frac{33267556681564784293908512651120282906448436}{1362760873221484918289678062664015532988759989} a^{13} - \frac{4781443710981371431649355671770520348886339}{151417874802387213143297562518223948109862221} a^{12} + \frac{13785127800531364704061338640036304074702118}{1362760873221484918289678062664015532988759989} a^{11} - \frac{262284760809363260222923603790392207349480197}{1362760873221484918289678062664015532988759989} a^{10} - \frac{22675163741653209936980896239761068879856}{227013305550805417006443122216227808260663} a^{9} + \frac{17214319167168021696761353921502091872461127}{1362760873221484918289678062664015532988759989} a^{8} + \frac{84165454458158800415038765012061253556062553}{454253624407161639429892687554671844329586663} a^{7} + \frac{312262126665253290966104193710122228847565184}{1362760873221484918289678062664015532988759989} a^{6} - \frac{517145674484453517550401366406083332224389355}{1362760873221484918289678062664015532988759989} a^{5} - \frac{153188423168955194392112361916923498055434346}{1362760873221484918289678062664015532988759989} a^{4} + \frac{314097875675498316502103467187543262895974233}{1362760873221484918289678062664015532988759989} a^{3} + \frac{206284356829204019807606476872394818051372893}{454253624407161639429892687554671844329586663} a^{2} - \frac{19479075382066727311139230551823918571934242}{1362760873221484918289678062664015532988759989} a + \frac{19301539846942579792258277997326458860504955}{151417874802387213143297562518223948109862221}$

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:  $9$
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:  \( 318354084.845 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1461:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1461 are not computed
Character table for t16n1461 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), 4.4.7168.1 x2, 4.4.25088.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
137Data not computed