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Properties

Label 16.8.73924058337...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{52}\cdot 3^{6}\cdot 5^{12}\cdot 7^{6}\cdot 13^{8}\cdot 31^{2}$
Root discriminant $551.82$
Ramified primes $2, 3, 5, 7, 13, 31$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T1697

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![586846391481, 0, 445778900028, 0, 38690866830, 0, -428011752, 0, -74050561, 0, 670992, 0, 26810, 0, -348, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 348*x^14 + 26810*x^12 + 670992*x^10 - 74050561*x^8 - 428011752*x^6 + 38690866830*x^4 + 445778900028*x^2 + 586846391481)
 
gp: K = bnfinit(x^16 - 348*x^14 + 26810*x^12 + 670992*x^10 - 74050561*x^8 - 428011752*x^6 + 38690866830*x^4 + 445778900028*x^2 + 586846391481, 1)
 

Normalized defining polynomial

\( x^{16} - 348 x^{14} + 26810 x^{12} + 670992 x^{10} - 74050561 x^{8} - 428011752 x^{6} + 38690866830 x^{4} + 445778900028 x^{2} + 586846391481 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(73924058337595330806214217957376000000000000=2^{52}\cdot 3^{6}\cdot 5^{12}\cdot 7^{6}\cdot 13^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $551.82$
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:  $2, 3, 5, 7, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{28} a^{10} - \frac{1}{4} a^{9} - \frac{3}{28} a^{8} - \frac{1}{2} a^{7} - \frac{13}{28} a^{6} - \frac{1}{4} a^{5} - \frac{5}{28} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{28} a^{11} + \frac{1}{7} a^{9} - \frac{1}{4} a^{8} + \frac{1}{28} a^{7} - \frac{1}{2} a^{6} + \frac{1}{14} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{1092} a^{12} + \frac{1}{91} a^{10} - \frac{1}{4} a^{9} + \frac{173}{1092} a^{8} - \frac{1}{2} a^{7} - \frac{87}{182} a^{6} - \frac{1}{4} a^{5} + \frac{317}{1092} a^{4} + \frac{5}{26} a^{2} - \frac{1}{4} a - \frac{6}{13}$, $\frac{1}{1092} a^{13} + \frac{1}{91} a^{11} - \frac{25}{273} a^{9} - \frac{1}{4} a^{8} + \frac{2}{91} a^{7} - \frac{1}{2} a^{6} + \frac{11}{273} a^{5} - \frac{1}{4} a^{4} + \frac{5}{26} a^{3} + \frac{15}{52} a - \frac{1}{4}$, $\frac{1}{10562058268895828862853805797330306836} a^{14} - \frac{1153592477785202493101923529785389}{3520686089631942954284601932443435612} a^{12} - \frac{164640505590249519174640161577728127}{10562058268895828862853805797330306836} a^{10} - \frac{1}{4} a^{9} + \frac{77354462413110333073998577858880377}{880171522407985738571150483110858903} a^{8} - \frac{1}{2} a^{7} - \frac{45100962789205826496400080349845385}{1508865466985118408979115113904329548} a^{6} - \frac{1}{4} a^{5} + \frac{55678269975873363357493982564137318}{125738788915426534081592926158694129} a^{4} - \frac{8476342833105885157283647252814913}{251477577830853068163185852317388258} a^{2} - \frac{1}{4} a + \frac{215075914916600023861903714828152317}{502955155661706136326371704634776516}$, $\frac{1}{385293323591050941088043981680812263070444} a^{15} + \frac{110046996167160729375625641447381551507}{385293323591050941088043981680812263070444} a^{13} - \frac{94749028759221799905251997293756924687}{27520951685075067220574570120058018790746} a^{11} - \frac{16271189569788753260457452822724322572557}{385293323591050941088043981680812263070444} a^{9} - \frac{7998306363219732020323829903278162873981}{96323330897762735272010995420203065767611} a^{7} + \frac{141588961674866707754317688706041401007029}{385293323591050941088043981680812263070444} a^{5} - \frac{4173877361875122187059843568693539182249}{18347301123383378147049713413372012527164} a^{3} + \frac{1302124871216997700885236192948081189806}{4586825280845844536762428353343003131791} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  $11$
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:  \( 679954104152000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1697:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 152 conjugacy class representatives for t16n1697 are not computed
Character table for t16n1697 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.6397254549504000000.1

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$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}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$