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Properties

Label 16.8.33531062442...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{36}\cdot 5^{12}\cdot 29^{4}\cdot 41^{4}$
Root discriminant $93.40$
Ramified primes $2, 5, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3611349436, -7573198816, 6195469544, -1951919440, 330768, 202341208, -65404272, 2909352, 2400752, -487616, 29412, 3640, -580, 276, -60, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 60*x^14 + 276*x^13 - 580*x^12 + 3640*x^11 + 29412*x^10 - 487616*x^9 + 2400752*x^8 + 2909352*x^7 - 65404272*x^6 + 202341208*x^5 + 330768*x^4 - 1951919440*x^3 + 6195469544*x^2 - 7573198816*x + 3611349436)
 
gp: K = bnfinit(x^16 - 4*x^15 - 60*x^14 + 276*x^13 - 580*x^12 + 3640*x^11 + 29412*x^10 - 487616*x^9 + 2400752*x^8 + 2909352*x^7 - 65404272*x^6 + 202341208*x^5 + 330768*x^4 - 1951919440*x^3 + 6195469544*x^2 - 7573198816*x + 3611349436, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 60 x^{14} + 276 x^{13} - 580 x^{12} + 3640 x^{11} + 29412 x^{10} - 487616 x^{9} + 2400752 x^{8} + 2909352 x^{7} - 65404272 x^{6} + 202341208 x^{5} + 330768 x^{4} - 1951919440 x^{3} + 6195469544 x^{2} - 7573198816 x + 3611349436 \)

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:  \(33531062442740678656000000000000=2^{36}\cdot 5^{12}\cdot 29^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.40$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{60} a^{12} + \frac{1}{20} a^{11} + \frac{1}{30} a^{10} - \frac{1}{15} a^{9} + \frac{7}{60} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{4}{15} a^{5} + \frac{3}{10} a^{4} + \frac{11}{30} a^{3} + \frac{4}{15} a^{2} - \frac{7}{15} a - \frac{7}{30}$, $\frac{1}{60} a^{13} - \frac{7}{60} a^{11} + \frac{1}{12} a^{10} + \frac{1}{15} a^{9} - \frac{1}{20} a^{8} + \frac{1}{5} a^{7} + \frac{13}{30} a^{6} + \frac{1}{10} a^{5} + \frac{7}{15} a^{4} + \frac{1}{6} a^{3} + \frac{7}{30} a^{2} - \frac{1}{3} a - \frac{3}{10}$, $\frac{1}{60} a^{14} - \frac{1}{15} a^{11} + \frac{1}{20} a^{10} - \frac{1}{60} a^{9} + \frac{1}{60} a^{8} + \frac{1}{30} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{4}{15} a^{4} - \frac{1}{5} a^{3} + \frac{1}{30} a^{2} + \frac{13}{30} a + \frac{11}{30}$, $\frac{1}{461346427268492111217515410285394035458900544378966976300780460} a^{15} - \frac{439882811682396326432335553525767230800483592682022030041041}{153782142422830703739171803428464678486300181459655658766926820} a^{14} - \frac{1151480521131127285179146326135488257783940435022635585198859}{230673213634246055608757705142697017729450272189483488150390230} a^{13} + \frac{2982720251209695307788626212854503872046359276241796719940317}{461346427268492111217515410285394035458900544378966976300780460} a^{12} - \frac{13063887930372535541126855305745865317612330904100850057144303}{230673213634246055608757705142697017729450272189483488150390230} a^{11} - \frac{14050954148746743640785411910268548174869575987501958069888803}{461346427268492111217515410285394035458900544378966976300780460} a^{10} + \frac{18880896878588147080571699022031536410072956455099740690511233}{461346427268492111217515410285394035458900544378966976300780460} a^{9} - \frac{3339297498162389544450921000979210480091062689874828491680459}{92269285453698422243503082057078807091780108875793395260156092} a^{8} + \frac{957848126631218951980754670161007639912327797694069750781147}{38445535605707675934792950857116169621575045364913914691731705} a^{7} + \frac{5053386903834569062440721709095357153902004378615880705802004}{23067321363424605560875770514269701772945027218948348815039023} a^{6} - \frac{50147056176902753029908889705567142233935014109394561237474739}{115336606817123027804378852571348508864725136094741744075195115} a^{5} - \frac{8428881996573048705204538662209756592328032812096293304085703}{230673213634246055608757705142697017729450272189483488150390230} a^{4} - \frac{2013342415955244955382104824551346439152887738853490546979863}{7689107121141535186958590171423233924315009072982782938346341} a^{3} + \frac{38119792296070126712248334750529687810238512897889998397312667}{230673213634246055608757705142697017729450272189483488150390230} a^{2} + \frac{52335685563693545309908818678667415527147575815282202137966083}{230673213634246055608757705142697017729450272189483488150390230} a - \frac{11065509851631702843571224612494766316505049737697554109731111}{76891071211415351869585901714232339243150090729827829383463410}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 1678641423.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$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}$
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}$
41Data not computed