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Properties

Label 16.0.64803441755...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 61^{12}$
Root discriminant $72.98$
Ramified primes $5, 61$
Class number $360$ (GRH)
Class group $[2, 6, 30]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63101531, -33563693, 37293974, -17027763, 8645703, -1876433, 983391, -207249, 112208, -20673, 12609, -1964, 978, -126, 41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 41*x^14 - 126*x^13 + 978*x^12 - 1964*x^11 + 12609*x^10 - 20673*x^9 + 112208*x^8 - 207249*x^7 + 983391*x^6 - 1876433*x^5 + 8645703*x^4 - 17027763*x^3 + 37293974*x^2 - 33563693*x + 63101531)
 
gp: K = bnfinit(x^16 - 4*x^15 + 41*x^14 - 126*x^13 + 978*x^12 - 1964*x^11 + 12609*x^10 - 20673*x^9 + 112208*x^8 - 207249*x^7 + 983391*x^6 - 1876433*x^5 + 8645703*x^4 - 17027763*x^3 + 37293974*x^2 - 33563693*x + 63101531, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 41 x^{14} - 126 x^{13} + 978 x^{12} - 1964 x^{11} + 12609 x^{10} - 20673 x^{9} + 112208 x^{8} - 207249 x^{7} + 983391 x^{6} - 1876433 x^{5} + 8645703 x^{4} - 17027763 x^{3} + 37293974 x^{2} - 33563693 x + 63101531 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(648034417553121620683837890625=5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.98$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{30} a^{8} - \frac{1}{15} a^{7} - \frac{7}{15} a^{6} - \frac{1}{5} a^{5} - \frac{1}{30} a^{4} + \frac{1}{15} a^{3} + \frac{3}{10} a^{2} + \frac{11}{30} a + \frac{11}{30}$, $\frac{1}{30} a^{9} + \frac{2}{5} a^{7} - \frac{2}{15} a^{6} - \frac{13}{30} a^{5} + \frac{13}{30} a^{3} - \frac{1}{30} a^{2} + \frac{1}{10} a - \frac{4}{15}$, $\frac{1}{30} a^{10} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{2}{5} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{2}{5}$, $\frac{1}{30} a^{11} - \frac{1}{2} a^{7} - \frac{4}{15} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{4}{15} a - \frac{1}{3}$, $\frac{1}{290520} a^{12} - \frac{1}{96840} a^{11} - \frac{3491}{290520} a^{10} + \frac{2117}{290520} a^{9} - \frac{2}{4035} a^{8} - \frac{19957}{96840} a^{7} + \frac{29803}{72630} a^{6} + \frac{56261}{290520} a^{5} + \frac{11297}{24210} a^{4} + \frac{144113}{290520} a^{3} - \frac{799}{2152} a^{2} + \frac{2519}{24210} a - \frac{140053}{290520}$, $\frac{1}{290520} a^{13} - \frac{175}{14526} a^{11} + \frac{166}{36315} a^{10} - \frac{1159}{96840} a^{9} - \frac{733}{96840} a^{8} - \frac{99137}{290520} a^{7} - \frac{21883}{290520} a^{6} - \frac{30899}{96840} a^{5} - \frac{136759}{290520} a^{4} + \frac{2423}{9684} a^{3} + \frac{31331}{96840} a^{2} + \frac{76523}{290520} a - \frac{36757}{96840}$, $\frac{1}{1452600} a^{14} - \frac{1}{1452600} a^{13} + \frac{1003}{363150} a^{11} - \frac{1309}{161400} a^{10} + \frac{6101}{726300} a^{9} + \frac{9419}{726300} a^{8} - \frac{50021}{145260} a^{7} - \frac{2477}{80700} a^{6} - \frac{29939}{80700} a^{5} + \frac{128713}{1452600} a^{4} - \frac{642533}{1452600} a^{3} - \frac{59527}{145260} a^{2} - \frac{262717}{726300} a - \frac{457}{5400}$, $\frac{1}{28302823790626939035027454182600} a^{15} + \frac{326146177997678055321419}{2358568649218911586252287848550} a^{14} - \frac{39843116860108461706254739}{28302823790626939035027454182600} a^{13} + \frac{25384060960651270183491947}{28302823790626939035027454182600} a^{12} + \frac{53844221268252889587351746921}{14151411895313469517513727091300} a^{11} - \frac{9416550106576214924329576663}{7075705947656734758756863545650} a^{10} + \frac{181503268480063516621242012241}{28302823790626939035027454182600} a^{9} - \frac{112812665726769965843441414099}{14151411895313469517513727091300} a^{8} + \frac{9271719922291218793623245089849}{28302823790626939035027454182600} a^{7} + \frac{6841506978206559976736291367527}{14151411895313469517513727091300} a^{6} - \frac{13890918755530907427915108277}{471713729843782317250457569710} a^{5} - \frac{6644769897917824730479686292043}{14151411895313469517513727091300} a^{4} + \frac{1051748380618786587415541781947}{7075705947656734758756863545650} a^{3} + \frac{2220103757401965387381378307397}{9434274596875646345009151394200} a^{2} + \frac{1352457994609136379468203554871}{28302823790626939035027454182600} a - \frac{1012239656701527821143570536917}{4717137298437823172504575697100}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{30}$, which has order $360$ (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:  $7$
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:  \( 320579.979552 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), 4.0.5674525.1, \(\Q(\sqrt{5}, \sqrt{61})\), 4.0.226981.1, 4.4.7625.1 x2, 4.4.465125.1 x2, 8.0.32200233975625.2, 8.8.216341265625.1, 8.0.805005849390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$