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Properties

Label 16.16.1747511875...8976.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 17^{2}\cdot 16673^{6}$
Root discriminant $436.67$
Ramified primes $2, 17, 16673$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![726055567, 1636792368, -6503647516, 935572536, 3594006416, -581322312, -755140300, 76869440, 71711252, -3553552, -3180476, 56712, 62812, -344, -492, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 492*x^14 - 344*x^13 + 62812*x^12 + 56712*x^11 - 3180476*x^10 - 3553552*x^9 + 71711252*x^8 + 76869440*x^7 - 755140300*x^6 - 581322312*x^5 + 3594006416*x^4 + 935572536*x^3 - 6503647516*x^2 + 1636792368*x + 726055567)
 
gp: K = bnfinit(x^16 - 492*x^14 - 344*x^13 + 62812*x^12 + 56712*x^11 - 3180476*x^10 - 3553552*x^9 + 71711252*x^8 + 76869440*x^7 - 755140300*x^6 - 581322312*x^5 + 3594006416*x^4 + 935572536*x^3 - 6503647516*x^2 + 1636792368*x + 726055567, 1)
 

Normalized defining polynomial

\( x^{16} - 492 x^{14} - 344 x^{13} + 62812 x^{12} + 56712 x^{11} - 3180476 x^{10} - 3553552 x^{9} + 71711252 x^{8} + 76869440 x^{7} - 755140300 x^{6} - 581322312 x^{5} + 3594006416 x^{4} + 935572536 x^{3} - 6503647516 x^{2} + 1636792368 x + 726055567 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1747511875136079735717765427007261010558976=2^{48}\cdot 17^{2}\cdot 16673^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $436.67$
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:  $2, 17, 16673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{7}{16}$, $\frac{1}{272} a^{13} + \frac{7}{272} a^{12} - \frac{1}{34} a^{11} - \frac{3}{34} a^{10} + \frac{21}{272} a^{9} + \frac{43}{272} a^{8} - \frac{4}{17} a^{7} - \frac{8}{17} a^{6} + \frac{129}{272} a^{5} - \frac{41}{272} a^{4} + \frac{5}{34} a^{3} + \frac{9}{34} a^{2} + \frac{73}{272} a + \frac{7}{16}$, $\frac{1}{544} a^{14} + \frac{11}{544} a^{12} + \frac{1}{17} a^{11} + \frac{53}{544} a^{10} - \frac{13}{68} a^{9} + \frac{111}{544} a^{8} - \frac{7}{17} a^{7} + \frac{209}{544} a^{6} - \frac{4}{17} a^{5} - \frac{149}{544} a^{4} - \frac{13}{34} a^{3} + \frac{249}{544} a^{2} + \frac{19}{68} a + \frac{11}{32}$, $\frac{1}{13389776399305910720960348287028899401221186609822034435795385792} a^{15} + \frac{5775681269392802505255052500525089832388676617306906166147519}{13389776399305910720960348287028899401221186609822034435795385792} a^{14} + \frac{8094674060535954008590957329457258837271084741451128749485965}{13389776399305910720960348287028899401221186609822034435795385792} a^{13} - \frac{284355089187207395283152651653729688375249723088114306193753329}{13389776399305910720960348287028899401221186609822034435795385792} a^{12} + \frac{45205455788165634997879692197877485187161191285905202179935631}{1912825199900844388708621183861271343031598087117433490827912256} a^{11} - \frac{1543881926472971953447231710238695599428638120833207157709082073}{13389776399305910720960348287028899401221186609822034435795385792} a^{10} + \frac{526093910838215645144876021393694153800873510121984982576857085}{13389776399305910720960348287028899401221186609822034435795385792} a^{9} + \frac{2750264780512823615855365053121254002361769078998002786800624079}{13389776399305910720960348287028899401221186609822034435795385792} a^{8} + \frac{2458075867831291347752882438687663927641146655693594131504818249}{13389776399305910720960348287028899401221186609822034435795385792} a^{7} - \frac{4457369281001209357995668980020275194312737669053643415284019033}{13389776399305910720960348287028899401221186609822034435795385792} a^{6} - \frac{5931990326869640087323221564019858796550521571263810672192687563}{13389776399305910720960348287028899401221186609822034435795385792} a^{5} - \frac{5089810534826193518557034728581498073689497711884716345799662329}{13389776399305910720960348287028899401221186609822034435795385792} a^{4} - \frac{4959890360264236336357749466399460952877067507651972307801621195}{13389776399305910720960348287028899401221186609822034435795385792} a^{3} + \frac{948399778685264421196681615213409090749746044642148611426684845}{1912825199900844388708621183861271343031598087117433490827912256} a^{2} - \frac{64044827479313663803899356724050399131577177620946238770300711}{13389776399305910720960348287028899401221186609822034435795385792} a + \frac{39397876693236949559673709277874015442611022944585884544498403}{787633905841524160056491075707582317718893329989531437399728576}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 28831028658100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.1067072.2, 4.4.34146304.1, 8.8.1165970076860416.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
16673Data not computed