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Properties

Label 16.0.12374161428...9069.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 61^{7}$
Root discriminant $56.99$
Ramified primes $13, 61$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68877, 39825, 100470, 40321, -25968, -42540, 71872, -48033, 24026, -9143, 3633, -1073, 330, -74, 20, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 20*x^14 - 74*x^13 + 330*x^12 - 1073*x^11 + 3633*x^10 - 9143*x^9 + 24026*x^8 - 48033*x^7 + 71872*x^6 - 42540*x^5 - 25968*x^4 + 40321*x^3 + 100470*x^2 + 39825*x + 68877)
 
gp: K = bnfinit(x^16 - x^15 + 20*x^14 - 74*x^13 + 330*x^12 - 1073*x^11 + 3633*x^10 - 9143*x^9 + 24026*x^8 - 48033*x^7 + 71872*x^6 - 42540*x^5 - 25968*x^4 + 40321*x^3 + 100470*x^2 + 39825*x + 68877, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 20 x^{14} - 74 x^{13} + 330 x^{12} - 1073 x^{11} + 3633 x^{10} - 9143 x^{9} + 24026 x^{8} - 48033 x^{7} + 71872 x^{6} - 42540 x^{5} - 25968 x^{4} + 40321 x^{3} + 100470 x^{2} + 39825 x + 68877 \)

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:  \(12374161428874698259143289069=13^{14}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.99$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{2}{9} a^{9} + \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{2785274765525232777404959576866433843821} a^{15} - \frac{40208067082534064924627264684217461777}{2785274765525232777404959576866433843821} a^{14} - \frac{1822716770740508324449764448594573469}{2785274765525232777404959576866433843821} a^{13} + \frac{137502345272093078532042068079671789096}{928424921841744259134986525622144614607} a^{12} - \frac{51541503617651668269903735838246051303}{309474973947248086378328841874048204869} a^{11} - \frac{101467582705843287834061914874248173348}{2785274765525232777404959576866433843821} a^{10} + \frac{270046609846101998459896063483066492973}{2785274765525232777404959576866433843821} a^{9} + \frac{1194690079912804039775688145746233875646}{2785274765525232777404959576866433843821} a^{8} - \frac{126735958615291270043790554251629502963}{928424921841744259134986525622144614607} a^{7} - \frac{405053177475099687074375912056991246879}{928424921841744259134986525622144614607} a^{6} - \frac{663744873387337975044766849318642558634}{2785274765525232777404959576866433843821} a^{5} + \frac{1105884965489243857844332289557698200342}{2785274765525232777404959576866433843821} a^{4} - \frac{47834540310671289041787624542648512526}{2785274765525232777404959576866433843821} a^{3} + \frac{100604445786942669882199085256774359206}{928424921841744259134986525622144614607} a^{2} - \frac{46113767460391631244349289098497825353}{103158324649082695459442947291349401623} a - \frac{3104101637609884794176152567335814409}{103158324649082695459442947291349401623}$

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

Class group and class number

$C_{2}\times C_{4}$, 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:  \( -1 \) (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:  \( 40233575.6332 \) (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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.17960556289.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$