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Properties

Label 16.4.24109722907...7601.8
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11651, 59010, 78008, 22319, 10911, 14278, 579, -10209, -7953, 9, 1693, 233, -133, 2, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 13*x^14 + 2*x^13 - 133*x^12 + 233*x^11 + 1693*x^10 + 9*x^9 - 7953*x^8 - 10209*x^7 + 579*x^6 + 14278*x^5 + 10911*x^4 + 22319*x^3 + 78008*x^2 + 59010*x - 11651)
 
gp: K = bnfinit(x^16 - x^15 + 13*x^14 + 2*x^13 - 133*x^12 + 233*x^11 + 1693*x^10 + 9*x^9 - 7953*x^8 - 10209*x^7 + 579*x^6 + 14278*x^5 + 10911*x^4 + 22319*x^3 + 78008*x^2 + 59010*x - 11651, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 13 x^{14} + 2 x^{13} - 133 x^{12} + 233 x^{11} + 1693 x^{10} + 9 x^{9} - 7953 x^{8} - 10209 x^{7} + 579 x^{6} + 14278 x^{5} + 10911 x^{4} + 22319 x^{3} + 78008 x^{2} + 59010 x - 11651 \)

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:  \(24109722907876309716269637601=13^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.41$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{4}{13} a^{11} - \frac{6}{13} a^{10} + \frac{5}{13} a^{9} - \frac{1}{13} a^{7} + \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{3} - \frac{2}{13} a^{2} + \frac{5}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{13} + \frac{4}{13} a^{11} - \frac{6}{13} a^{10} - \frac{6}{13} a^{9} - \frac{1}{13} a^{8} - \frac{2}{13} a^{7} - \frac{3}{13} a^{6} + \frac{4}{13} a^{5} - \frac{1}{13} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} - \frac{2}{13} a + \frac{3}{13}$, $\frac{1}{663} a^{14} - \frac{1}{663} a^{13} + \frac{3}{221} a^{12} - \frac{49}{221} a^{11} + \frac{35}{663} a^{10} + \frac{56}{663} a^{9} - \frac{79}{663} a^{8} - \frac{266}{663} a^{7} - \frac{191}{663} a^{6} + \frac{239}{663} a^{5} + \frac{1}{3} a^{4} - \frac{142}{663} a^{3} + \frac{82}{663} a^{2} - \frac{55}{221} a + \frac{160}{663}$, $\frac{1}{31107341787668830960715417747995551} a^{15} + \frac{5872780804260655905153200053358}{10369113929222943653571805915998517} a^{14} + \frac{1193383228177666243811668210380362}{31107341787668830960715417747995551} a^{13} + \frac{166954373292971306845744142931838}{10369113929222943653571805915998517} a^{12} + \frac{516604430421148529884756399143200}{2392872445205294689285801365230427} a^{11} - \frac{7170032534249580621853591116968348}{31107341787668830960715417747995551} a^{10} + \frac{7201195388542538533399639669825480}{31107341787668830960715417747995551} a^{9} - \frac{400456139292914034101440423682859}{10369113929222943653571805915998517} a^{8} + \frac{12916189043200074472749610379466617}{31107341787668830960715417747995551} a^{7} + \frac{1105894382871595356744260656081422}{10369113929222943653571805915998517} a^{6} + \frac{8753758697719452443549203716768010}{31107341787668830960715417747995551} a^{5} + \frac{3551948240789817057184009412715535}{31107341787668830960715417747995551} a^{4} - \frac{584324290108378375278162595762769}{10369113929222943653571805915998517} a^{3} - \frac{813547524430770045087433534919834}{31107341787668830960715417747995551} a^{2} - \frac{5798915949985581588534962679282947}{31107341787668830960715417747995551} a + \frac{3222899702489178385353777723891529}{31107341787668830960715417747995551}$

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:  \( 81561758.266 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$