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Properties

Label 16.0.26175179494...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{7}$
Root discriminant $25.18$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![341, -1329, 3173, -5494, 7181, -7507, 6405, -3640, 1711, 115, -285, 326, -46, 11, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 + 11*x^13 - 46*x^12 + 326*x^11 - 285*x^10 + 115*x^9 + 1711*x^8 - 3640*x^7 + 6405*x^6 - 7507*x^5 + 7181*x^4 - 5494*x^3 + 3173*x^2 - 1329*x + 341)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 + 11*x^13 - 46*x^12 + 326*x^11 - 285*x^10 + 115*x^9 + 1711*x^8 - 3640*x^7 + 6405*x^6 - 7507*x^5 + 7181*x^4 - 5494*x^3 + 3173*x^2 - 1329*x + 341, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} + 11 x^{13} - 46 x^{12} + 326 x^{11} - 285 x^{10} + 115 x^{9} + 1711 x^{8} - 3640 x^{7} + 6405 x^{6} - 7507 x^{5} + 7181 x^{4} - 5494 x^{3} + 3173 x^{2} - 1329 x + 341 \)

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:  \(26175179494800048828125=5^{12}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{5}{11} a^{11} - \frac{5}{11} a^{10} + \frac{2}{11} a^{9} + \frac{2}{11} a^{8} + \frac{5}{11} a^{7} + \frac{5}{11} a^{6} + \frac{1}{11} a^{5} + \frac{2}{11} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{242} a^{13} - \frac{7}{242} a^{12} - \frac{39}{242} a^{11} - \frac{27}{121} a^{10} - \frac{23}{121} a^{9} + \frac{111}{242} a^{8} + \frac{58}{121} a^{7} + \frac{12}{121} a^{6} - \frac{12}{121} a^{5} + \frac{45}{121} a^{4} - \frac{3}{22} a^{3} + \frac{16}{121} a^{2} - \frac{29}{121} a + \frac{3}{22}$, $\frac{1}{5324} a^{14} + \frac{1}{2662} a^{13} + \frac{35}{1331} a^{12} + \frac{805}{5324} a^{11} - \frac{871}{2662} a^{10} + \frac{2359}{5324} a^{9} + \frac{2325}{5324} a^{8} - \frac{1039}{2662} a^{7} - \frac{194}{1331} a^{6} + \frac{179}{2662} a^{5} + \frac{2229}{5324} a^{4} + \frac{703}{5324} a^{3} - \frac{369}{2662} a^{2} - \frac{731}{5324} a - \frac{83}{484}$, $\frac{1}{2844035905368519928} a^{15} + \frac{331499990171}{23504428969987768} a^{14} + \frac{1092877305633357}{1422017952684259964} a^{13} + \frac{104774337812059605}{2844035905368519928} a^{12} - \frac{813424902993018357}{2844035905368519928} a^{11} - \frac{1388036865968997143}{2844035905368519928} a^{10} - \frac{156079737400427421}{355504488171064991} a^{9} - \frac{622543451771251721}{2844035905368519928} a^{8} - \frac{53571983435151361}{1422017952684259964} a^{7} - \frac{257998241164504469}{1422017952684259964} a^{6} + \frac{902148083200375875}{2844035905368519928} a^{5} + \frac{247198693033672159}{711008976342129982} a^{4} - \frac{33409599372810275}{2844035905368519928} a^{3} + \frac{1300142849079734679}{2844035905368519928} a^{2} + \frac{89983425286846876}{355504488171064991} a + \frac{89179580615253393}{258548718669865448}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{1271792410331283}{711008976342129982} a^{15} - \frac{172905323038244}{32318589833733181} a^{14} + \frac{21417514104026907}{711008976342129982} a^{13} - \frac{1296837199238055}{355504488171064991} a^{12} - \frac{76389626470768101}{711008976342129982} a^{11} + \frac{495599169325611581}{711008976342129982} a^{10} - \frac{708098693311952841}{711008976342129982} a^{9} + \frac{396468983550854091}{711008976342129982} a^{8} + \frac{1291591836140363430}{355504488171064991} a^{7} - \frac{3313580981532564544}{355504488171064991} a^{6} + \frac{12148718143417287077}{711008976342129982} a^{5} - \frac{14004272631238861057}{711008976342129982} a^{4} + \frac{14595353832705842003}{711008976342129982} a^{3} - \frac{10421789426915608955}{711008976342129982} a^{2} + \frac{6219638508380286237}{711008976342129982} a - \frac{199547318904810971}{64637179667466362} \) (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:  \( 80758.4330938 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

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 t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.159390625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$