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Properties

Label 16.0.66092328224...625.11
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{8}$
Root discriminant $41.09$
Ramified primes $5, 101$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![510242801, -59152745, -238936324, 16695220, 49799537, -149590, -6343597, -462345, 566450, 75280, -36793, -5950, 1757, 250, -56, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 56*x^14 + 250*x^13 + 1757*x^12 - 5950*x^11 - 36793*x^10 + 75280*x^9 + 566450*x^8 - 462345*x^7 - 6343597*x^6 - 149590*x^5 + 49799537*x^4 + 16695220*x^3 - 238936324*x^2 - 59152745*x + 510242801)
 
gp: K = bnfinit(x^16 - 5*x^15 - 56*x^14 + 250*x^13 + 1757*x^12 - 5950*x^11 - 36793*x^10 + 75280*x^9 + 566450*x^8 - 462345*x^7 - 6343597*x^6 - 149590*x^5 + 49799537*x^4 + 16695220*x^3 - 238936324*x^2 - 59152745*x + 510242801, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 56 x^{14} + 250 x^{13} + 1757 x^{12} - 5950 x^{11} - 36793 x^{10} + 75280 x^{9} + 566450 x^{8} - 462345 x^{7} - 6343597 x^{6} - 149590 x^{5} + 49799537 x^{4} + 16695220 x^{3} - 238936324 x^{2} - 59152745 x + 510242801 \)

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:  \(66092328224370123291015625=5^{14}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.09$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{515252850394319845178332348244953165675821911739660229} a^{15} + \frac{187539751248849237733332521067615213190633435285147762}{515252850394319845178332348244953165675821911739660229} a^{14} + \frac{238909280507371404258631956435594776227565834332032279}{515252850394319845178332348244953165675821911739660229} a^{13} - \frac{146352373890660579178119751972843819985423734756194847}{515252850394319845178332348244953165675821911739660229} a^{12} + \frac{182281171844407676397502877379597552607528789361103068}{515252850394319845178332348244953165675821911739660229} a^{11} + \frac{139297077644537142948261564432979768097074416167400792}{515252850394319845178332348244953165675821911739660229} a^{10} - \frac{180533303703037664624079087903901788227851331853934711}{515252850394319845178332348244953165675821911739660229} a^{9} + \frac{3921225649708006451861634755330554888749857061607125}{515252850394319845178332348244953165675821911739660229} a^{8} - \frac{20923954839809379286454037759150548724617294910054085}{515252850394319845178332348244953165675821911739660229} a^{7} - \frac{18422215474628528247951136685996455976379297980793888}{515252850394319845178332348244953165675821911739660229} a^{6} + \frac{187246043395927932186864950168711651484517046179590280}{515252850394319845178332348244953165675821911739660229} a^{5} + \frac{97088196414669248520613728444803195833833557575392024}{515252850394319845178332348244953165675821911739660229} a^{4} - \frac{208417530384744115334032370000213138412498164885660381}{515252850394319845178332348244953165675821911739660229} a^{3} + \frac{38175982935493924489601423845011774839490428476569974}{515252850394319845178332348244953165675821911739660229} a^{2} + \frac{222616310403751379825945833009255413665907896134832291}{515252850394319845178332348244953165675821911739660229} a + \frac{24809951918484241300795275937492566103072639123628147}{515252850394319845178332348244953165675821911739660229}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( -\frac{1431748995101215615752797975685346756105530}{5237320726505319575714135333498878499667841470809} a^{15} + \frac{3612561819301964248390326022336162318187622}{5237320726505319575714135333498878499667841470809} a^{14} + \frac{87390146592103003982990408564722748792079882}{5237320726505319575714135333498878499667841470809} a^{13} - \frac{129650098064057666073886751750496840424969841}{5237320726505319575714135333498878499667841470809} a^{12} - \frac{2745446505678591041797137406743514975971990863}{5237320726505319575714135333498878499667841470809} a^{11} + \frac{1150011735684781325479790921924791853778866274}{5237320726505319575714135333498878499667841470809} a^{10} + \frac{52624229112889582946938250514278754622018746976}{5237320726505319575714135333498878499667841470809} a^{9} + \frac{35544829216824390089021112331534002859018257986}{5237320726505319575714135333498878499667841470809} a^{8} - \frac{660790872126300221343643259526820110911645537691}{5237320726505319575714135333498878499667841470809} a^{7} - \frac{1125687429113119615690077072790609735361384689088}{5237320726505319575714135333498878499667841470809} a^{6} + \frac{5339191691102188921540098187362496413483029918589}{5237320726505319575714135333498878499667841470809} a^{5} + \frac{14213395849864746540686093078500062639124003253817}{5237320726505319575714135333498878499667841470809} a^{4} - \frac{26198233075740479190914714023552357084327240629590}{5237320726505319575714135333498878499667841470809} a^{3} - \frac{87200197892318032427450502954444443916228885647773}{5237320726505319575714135333498878499667841470809} a^{2} + \frac{60241946472866514802239694220781904049009724958979}{5237320726505319575714135333498878499667841470809} a + \frac{212984413475243133612586940005034316185230592948727}{5237320726505319575714135333498878499667841470809} \) (order $10$)
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:  \( 872881.884776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
101Data not computed