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Properties

Label 16.4.53055503619...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 3^{6}\cdot 5^{8}\cdot 7^{2}\cdot 97^{4}$
Root discriminant $54.05$
Ramified primes $2, 3, 5, 7, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35721, 0, -30618, 0, -13149, 0, -24264, 0, -2731, 0, -92, 0, -41, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 14*x^14 - 41*x^12 - 92*x^10 - 2731*x^8 - 24264*x^6 - 13149*x^4 - 30618*x^2 + 35721)
 
gp: K = bnfinit(x^16 + 14*x^14 - 41*x^12 - 92*x^10 - 2731*x^8 - 24264*x^6 - 13149*x^4 - 30618*x^2 + 35721, 1)
 

Normalized defining polynomial

\( x^{16} + 14 x^{14} - 41 x^{12} - 92 x^{10} - 2731 x^{8} - 24264 x^{6} - 13149 x^{4} - 30618 x^{2} + 35721 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5305550361918544281600000000=2^{32}\cdot 3^{6}\cdot 5^{8}\cdot 7^{2}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.05$
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:  $2, 3, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{3}{10} a^{6} - \frac{1}{2} a^{4} - \frac{1}{10} a^{2} - \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{3}{10} a^{7} - \frac{1}{2} a^{5} - \frac{1}{10} a^{3} - \frac{1}{10} a$, $\frac{1}{30} a^{10} - \frac{1}{30} a^{8} - \frac{11}{30} a^{6} - \frac{11}{30} a^{4} - \frac{13}{30} a^{2} - \frac{2}{5}$, $\frac{1}{90} a^{11} - \frac{2}{45} a^{9} - \frac{16}{45} a^{7} + \frac{17}{45} a^{5} - \frac{4}{9} a^{3} - \frac{1}{10} a$, $\frac{1}{2700} a^{12} - \frac{1}{180} a^{11} - \frac{11}{1350} a^{10} + \frac{1}{45} a^{9} + \frac{49}{2700} a^{8} - \frac{29}{90} a^{7} - \frac{587}{2700} a^{6} + \frac{14}{45} a^{5} - \frac{1147}{2700} a^{4} + \frac{2}{9} a^{3} - \frac{11}{150} a^{2} + \frac{1}{20} a + \frac{49}{100}$, $\frac{1}{2700} a^{13} - \frac{7}{2700} a^{11} - \frac{11}{2700} a^{9} - \frac{1}{20} a^{8} + \frac{283}{2700} a^{7} - \frac{7}{20} a^{6} + \frac{713}{2700} a^{5} + \frac{1}{4} a^{4} - \frac{133}{450} a^{3} - \frac{9}{20} a^{2} + \frac{11}{25} a + \frac{1}{20}$, $\frac{1}{94470354423900} a^{14} + \frac{81522017}{13495764917700} a^{12} - \frac{228949726403}{94470354423900} a^{10} - \frac{1}{20} a^{9} + \frac{335799188299}{7266950340300} a^{8} - \frac{7}{20} a^{7} - \frac{884926509559}{94470354423900} a^{6} + \frac{1}{4} a^{5} + \frac{49483951073}{629802362826} a^{4} - \frac{9}{20} a^{3} + \frac{38276707127}{349890201570} a^{2} + \frac{1}{20} a - \frac{16650035029}{41653595425}$, $\frac{1}{283411063271700} a^{15} + \frac{1269988367}{10121823688275} a^{13} + \frac{143949434302}{70852765817925} a^{11} - \frac{1}{60} a^{10} - \frac{74347751039}{2180085102090} a^{9} - \frac{1}{30} a^{8} + \frac{1884306585661}{141705531635850} a^{7} + \frac{1}{3} a^{6} - \frac{8182996338091}{31490118141300} a^{5} + \frac{13}{30} a^{4} - \frac{928265109389}{5248353023550} a^{3} - \frac{7}{30} a^{2} - \frac{13325349489}{41653595425} a + \frac{1}{4}$

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:  $9$
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:  \( 25289623.6407 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.216783360000.1

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 R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$