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Properties

Label 16.0.31445540991...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 61^{6}$
Root discriminant $19.10$
Ramified primes $5, 61$
Class number $2$
Class group $[2]$
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, -311, -102, 698, 151, -1006, -305, 764, 544, -227, -285, 2, 91, -9, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 - 9*x^13 + 91*x^12 + 2*x^11 - 285*x^10 - 227*x^9 + 544*x^8 + 764*x^7 - 305*x^6 - 1006*x^5 + 151*x^4 + 698*x^3 - 102*x^2 - 311*x + 121)
 
gp: K = bnfinit(x^16 - 3*x^15 - 3*x^14 - 9*x^13 + 91*x^12 + 2*x^11 - 285*x^10 - 227*x^9 + 544*x^8 + 764*x^7 - 305*x^6 - 1006*x^5 + 151*x^4 + 698*x^3 - 102*x^2 - 311*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 3 x^{14} - 9 x^{13} + 91 x^{12} + 2 x^{11} - 285 x^{10} - 227 x^{9} + 544 x^{8} + 764 x^{7} - 305 x^{6} - 1006 x^{5} + 151 x^{4} + 698 x^{3} - 102 x^{2} - 311 x + 121 \)

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:  \(314455409918212890625=5^{14}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.10$
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 not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{54292202355854488916} a^{15} + \frac{2323667351229732561}{54292202355854488916} a^{14} + \frac{3207428441089837055}{27146101177927244458} a^{13} - \frac{4264903982296113049}{54292202355854488916} a^{12} - \frac{406048893335994355}{4935654759623135356} a^{11} + \frac{2126518947335691613}{27146101177927244458} a^{10} - \frac{10111706772226831347}{27146101177927244458} a^{9} - \frac{24562141620567660465}{54292202355854488916} a^{8} - \frac{5319323527034466251}{27146101177927244458} a^{7} - \frac{2685812247320933043}{54292202355854488916} a^{6} - \frac{4139076134588344186}{13573050588963622229} a^{5} - \frac{5015020016101850754}{13573050588963622229} a^{4} - \frac{3331283005793455135}{27146101177927244458} a^{3} - \frac{2794388546182413657}{13573050588963622229} a^{2} + \frac{15753780065346350129}{54292202355854488916} a - \frac{635040607339563879}{2467827379811567678}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{190284878779365}{12216967226789939} a^{15} - \frac{447782149996330}{12216967226789939} a^{14} - \frac{783302195792377}{12216967226789939} a^{13} - \frac{10017545001523833}{48867868907159756} a^{12} + \frac{31497290043071043}{24433934453579878} a^{11} + \frac{38430943426881155}{48867868907159756} a^{10} - \frac{40616548243444039}{12216967226789939} a^{9} - \frac{150714944466639605}{24433934453579878} a^{8} + \frac{44002356233439341}{12216967226789939} a^{7} + \frac{331435870350701809}{24433934453579878} a^{6} + \frac{301024832478713067}{48867868907159756} a^{5} - \frac{483214551810821631}{48867868907159756} a^{4} - \frac{230750227237956575}{48867868907159756} a^{3} + \frac{81573381991782696}{12216967226789939} a^{2} + \frac{168437426012480055}{48867868907159756} a - \frac{139260534025267271}{48867868907159756} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19915.4165495 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.7625.1, \(\Q(\zeta_{5})\), 4.0.1525.1, 8.8.17732890625.1, 8.0.17732890625.1, 8.0.58140625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$