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Properties

Label 16.8.73904015491...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $73.59$
Ramified primes $2, 5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3590449, 2961393, -1483482, -2508605, -679622, 748264, 403748, -83954, -67991, 3734, 7296, -1028, -622, 125, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 125*x^13 - 622*x^12 - 1028*x^11 + 7296*x^10 + 3734*x^9 - 67991*x^8 - 83954*x^7 + 403748*x^6 + 748264*x^5 - 679622*x^4 - 2508605*x^3 - 1483482*x^2 + 2961393*x + 3590449)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 125*x^13 - 622*x^12 - 1028*x^11 + 7296*x^10 + 3734*x^9 - 67991*x^8 - 83954*x^7 + 403748*x^6 + 748264*x^5 - 679622*x^4 - 2508605*x^3 - 1483482*x^2 + 2961393*x + 3590449, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 125 x^{13} - 622 x^{12} - 1028 x^{11} + 7296 x^{10} + 3734 x^{9} - 67991 x^{8} - 83954 x^{7} + 403748 x^{6} + 748264 x^{5} - 679622 x^{4} - 2508605 x^{3} - 1483482 x^{2} + 2961393 x + 3590449 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(739040154910723253702500000000=2^{8}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.59$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2929030920252813858845721993491737216801176665294} a^{15} - \frac{531221193049613447621290779144937747464659798759}{2929030920252813858845721993491737216801176665294} a^{14} + \frac{4841973999905614960124011992564841977622949797}{27632367172196357158921905598978652988690345899} a^{13} + \frac{137375428891981825716686183376696887632401232101}{1464515460126406929422860996745868608400588332647} a^{12} - \frac{448853600057234996960557476354000232656583526353}{1464515460126406929422860996745868608400588332647} a^{11} - \frac{500754727855341025270479042599940267203821129128}{1464515460126406929422860996745868608400588332647} a^{10} + \frac{1153832673605130490382723389132325366739795987423}{2929030920252813858845721993491737216801176665294} a^{9} - \frac{568194260006902506573808653247977051564331919821}{1464515460126406929422860996745868608400588332647} a^{8} + \frac{319187518763170392965904830388936628393574942316}{1464515460126406929422860996745868608400588332647} a^{7} + \frac{634237808332532266192205732372750766439360936429}{1464515460126406929422860996745868608400588332647} a^{6} + \frac{377687572674073059210276871903819325119872208365}{1464515460126406929422860996745868608400588332647} a^{5} + \frac{1228314673841599479354269478881242306617284337071}{2929030920252813858845721993491737216801176665294} a^{4} - \frac{106032332808200766756643950179008219010838826536}{1464515460126406929422860996745868608400588332647} a^{3} + \frac{496505310699722875177001254306202669855699633326}{1464515460126406929422860996745868608400588332647} a^{2} + \frac{491886317908243943934853404033223047955707320142}{1464515460126406929422860996745868608400588332647} a + \frac{15970794458682550021488430185917072381039791705}{154159522118569150465564315446933537726377719226}$

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:  $11$
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:  \( 1017537675.62 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.4.2225.1, 4.4.64525.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$