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Properties

Label 16.0.74712195802...6224.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 7^{2}\cdot 23^{2}$
Root discriminant $23.29$
Ramified primes $2, 7, 23$
Class number $1$
Class group Trivial
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -168, 528, -904, 1744, -1920, 1896, -1376, 842, -264, 144, -88, 56, -16, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 16*x^13 + 56*x^12 - 88*x^11 + 144*x^10 - 264*x^9 + 842*x^8 - 1376*x^7 + 1896*x^6 - 1920*x^5 + 1744*x^4 - 904*x^3 + 528*x^2 - 168*x + 31)
 
gp: K = bnfinit(x^16 + 8*x^14 - 16*x^13 + 56*x^12 - 88*x^11 + 144*x^10 - 264*x^9 + 842*x^8 - 1376*x^7 + 1896*x^6 - 1920*x^5 + 1744*x^4 - 904*x^3 + 528*x^2 - 168*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 16 x^{13} + 56 x^{12} - 88 x^{11} + 144 x^{10} - 264 x^{9} + 842 x^{8} - 1376 x^{7} + 1896 x^{6} - 1920 x^{5} + 1744 x^{4} - 904 x^{3} + 528 x^{2} - 168 x + 31 \)

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:  \(7471219580228520116224=2^{58}\cdot 7^{2}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.29$
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:  $2, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{477261675570369883243807} a^{15} - \frac{141529996133157762651980}{477261675570369883243807} a^{14} + \frac{157478748241436325475926}{477261675570369883243807} a^{13} + \frac{197530460454173644065185}{477261675570369883243807} a^{12} + \frac{11892380795986922867749}{28074216210021757837871} a^{11} - \frac{128106869888052338836955}{477261675570369883243807} a^{10} - \frac{67524502657006453942975}{477261675570369883243807} a^{9} - \frac{156765715825011986065161}{477261675570369883243807} a^{8} - \frac{93567556972328152814669}{477261675570369883243807} a^{7} - \frac{111137246894223310476517}{477261675570369883243807} a^{6} + \frac{112128040047207403998626}{477261675570369883243807} a^{5} + \frac{55942810790441004895203}{477261675570369883243807} a^{4} + \frac{44809671435783710110675}{477261675570369883243807} a^{3} - \frac{126061869851864262339961}{477261675570369883243807} a^{2} - \frac{79262083303474983228492}{477261675570369883243807} a + \frac{49496185288868996107847}{477261675570369883243807}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 27253.6854245 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1024.1, \(\Q(\zeta_{16})^+\), 4.2.2048.1, 8.4.67108864.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$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}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$