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Properties

Label 16.4.11230910639...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{6}$
Root discriminant $367.83$
Ramified primes $2, 5, 71, 101$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25951599025, 0, 4458422800, 0, -1747851460, 0, -497016410, 0, -42903454, 0, -1539640, 0, -19303, 0, 50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 50*x^14 - 19303*x^12 - 1539640*x^10 - 42903454*x^8 - 497016410*x^6 - 1747851460*x^4 + 4458422800*x^2 + 25951599025)
 
gp: K = bnfinit(x^16 + 50*x^14 - 19303*x^12 - 1539640*x^10 - 42903454*x^8 - 497016410*x^6 - 1747851460*x^4 + 4458422800*x^2 + 25951599025, 1)
 

Normalized defining polynomial

\( x^{16} + 50 x^{14} - 19303 x^{12} - 1539640 x^{10} - 42903454 x^{8} - 497016410 x^{6} - 1747851460 x^{4} + 4458422800 x^{2} + 25951599025 \)

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:  \(112309106399498694797369730826240000000000=2^{24}\cdot 5^{10}\cdot 71^{8}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $367.83$
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:  $2, 5, 71, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{9} + \frac{1}{10} a^{7} + \frac{1}{5} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{60} a^{12} - \frac{1}{20} a^{11} - \frac{1}{60} a^{10} + \frac{3}{20} a^{9} - \frac{1}{12} a^{8} - \frac{3}{10} a^{7} + \frac{3}{20} a^{6} - \frac{1}{10} a^{5} - \frac{11}{60} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12}$, $\frac{1}{660} a^{13} - \frac{1}{30} a^{11} - \frac{38}{165} a^{9} - \frac{1}{4} a^{8} - \frac{49}{220} a^{7} - \frac{1}{4} a^{6} - \frac{53}{660} a^{5} - \frac{1}{2} a^{4} - \frac{31}{132} a^{3} - \frac{1}{2} a^{2} - \frac{61}{132} a + \frac{1}{4}$, $\frac{1}{13261756166835867615634170717300} a^{14} + \frac{680503442709902686194868}{467292324412821269049829835} a^{12} - \frac{4926194265570781279246919824}{1105146347236322301302847559775} a^{10} - \frac{1}{4} a^{9} + \frac{422842307109415438757781236327}{2652351233367173523126834143460} a^{8} - \frac{1}{4} a^{7} + \frac{3768650783412929691340935430891}{13261756166835867615634170717300} a^{6} - \frac{1}{2} a^{5} - \frac{128249580615699449949871216249}{530470246673434704625366828692} a^{4} - \frac{1}{2} a^{3} - \frac{11388853510346563715256414691}{26260903300665084387394397460} a^{2} + \frac{1}{4} a - \frac{14592807667772190418789313}{119367742275750383579065443}$, $\frac{1}{384590928838240160853390950801700} a^{15} - \frac{609296026959096199380900481}{894397508926139908961374304190} a^{13} - \frac{4229113285091449432626302246089}{192295464419120080426695475400850} a^{11} - \frac{1}{20} a^{10} - \frac{5142321889751473831127322346379}{25639395255882677390226063386780} a^{9} + \frac{3}{20} a^{8} + \frac{30533285956481680697438989060351}{384590928838240160853390950801700} a^{7} - \frac{3}{10} a^{6} - \frac{4412198693902679716091091211595}{15383637153529606434135638032068} a^{5} - \frac{1}{10} a^{4} + \frac{276126222735983560921519711}{2387354845515007671581308860} a^{3} + \frac{1}{4} a^{2} - \frac{4005905539023344065597949927}{12692769928654790787240625439} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( 27271857025800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{355}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{71}) \), 4.4.203656400.1, 4.4.2525.1, \(\Q(\sqrt{5}, \sqrt{71})\), 8.8.41475929260960000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71Data not computed
101Data not computed