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Properties

Label 16.16.5500525693...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 37^{14}$
Root discriminant $96.33$
Ramified primes $5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1831516, -4048152, -17141511, 58093288, -56521182, 12473518, 10976306, -5761286, -315387, 627708, -46611, -30236, 3788, 686, -104, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 104*x^14 + 686*x^13 + 3788*x^12 - 30236*x^11 - 46611*x^10 + 627708*x^9 - 315387*x^8 - 5761286*x^7 + 10976306*x^6 + 12473518*x^5 - 56521182*x^4 + 58093288*x^3 - 17141511*x^2 - 4048152*x + 1831516)
 
gp: K = bnfinit(x^16 - 6*x^15 - 104*x^14 + 686*x^13 + 3788*x^12 - 30236*x^11 - 46611*x^10 + 627708*x^9 - 315387*x^8 - 5761286*x^7 + 10976306*x^6 + 12473518*x^5 - 56521182*x^4 + 58093288*x^3 - 17141511*x^2 - 4048152*x + 1831516, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 104 x^{14} + 686 x^{13} + 3788 x^{12} - 30236 x^{11} - 46611 x^{10} + 627708 x^{9} - 315387 x^{8} - 5761286 x^{7} + 10976306 x^{6} + 12473518 x^{5} - 56521182 x^{4} + 58093288 x^{3} - 17141511 x^{2} - 4048152 x + 1831516 \)

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:  \(55005256933563283079156494140625=5^{14}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.33$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4}$, $\frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{3}{16} a^{6} - \frac{3}{16} a^{5} + \frac{7}{32} a^{4} - \frac{5}{16} a^{3} - \frac{3}{32} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{9} + \frac{7}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{16} a^{6} + \frac{13}{64} a^{5} + \frac{15}{64} a^{4} - \frac{25}{64} a^{3} + \frac{3}{64} a^{2} - \frac{5}{16} a + \frac{1}{16}$, $\frac{1}{512} a^{12} - \frac{1}{128} a^{11} - \frac{3}{128} a^{9} - \frac{11}{512} a^{8} + \frac{21}{256} a^{7} + \frac{41}{512} a^{6} + \frac{1}{8} a^{5} - \frac{49}{256} a^{4} + \frac{99}{256} a^{3} - \frac{97}{512} a^{2} - \frac{1}{4} a + \frac{17}{128}$, $\frac{1}{13312} a^{13} + \frac{9}{13312} a^{12} - \frac{1}{3328} a^{11} + \frac{41}{3328} a^{10} + \frac{73}{13312} a^{9} - \frac{1013}{13312} a^{8} + \frac{1387}{13312} a^{7} - \frac{2347}{13312} a^{6} + \frac{519}{6656} a^{5} + \frac{783}{3328} a^{4} - \frac{1539}{13312} a^{3} + \frac{1091}{13312} a^{2} - \frac{639}{3328} a - \frac{915}{3328}$, $\frac{1}{3700736} a^{14} - \frac{11}{925184} a^{13} - \frac{209}{284672} a^{12} + \frac{697}{462592} a^{11} + \frac{393}{284672} a^{10} - \frac{35825}{1850368} a^{9} - \frac{14533}{462592} a^{8} - \frac{39541}{1850368} a^{7} + \frac{185177}{3700736} a^{6} - \frac{135141}{1850368} a^{5} - \frac{879831}{3700736} a^{4} + \frac{24845}{1850368} a^{3} + \frac{1127041}{3700736} a^{2} - \frac{7997}{115648} a - \frac{271149}{925184}$, $\frac{1}{340589995685466227269197824} a^{15} + \frac{25785667989057563257}{340589995685466227269197824} a^{14} + \frac{3479786451751542356175}{340589995685466227269197824} a^{13} - \frac{316126444863973555418145}{340589995685466227269197824} a^{12} - \frac{2217359450240506094602435}{340589995685466227269197824} a^{11} + \frac{2331616918073927875726247}{340589995685466227269197824} a^{10} - \frac{1355794714026416165313829}{170294997842733113634598912} a^{9} - \frac{5887657453248990156996397}{170294997842733113634598912} a^{8} + \frac{2598124793943908214150747}{26199230437343555943784448} a^{7} - \frac{7498843138335816745650885}{340589995685466227269197824} a^{6} - \frac{3478589403248815694759101}{26199230437343555943784448} a^{5} - \frac{39515817636986719305875865}{340589995685466227269197824} a^{4} + \frac{14914815513345576955761131}{340589995685466227269197824} a^{3} - \frac{43821181757908690646291427}{340589995685466227269197824} a^{2} - \frac{11332145574262318429484397}{85147498921366556817299456} a + \frac{571765247225421156690529}{1199260548188261363623936}$

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:  \( 325484310652 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{185}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{37})\), 4.4.6331625.2, 4.4.6331625.1, 8.8.40089475140625.1, 8.8.7416552901015625.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
37Data not computed