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Properties

Label 16.0.61630862986...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 41^{6}$
Root discriminant $30.68$
Ramified primes $3, 5, 41$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
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![2416, -224, 1060, 5464, 5763, 5077, 4624, 1248, 706, -339, -88, -192, 20, -19, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 14*x^14 - 19*x^13 + 20*x^12 - 192*x^11 - 88*x^10 - 339*x^9 + 706*x^8 + 1248*x^7 + 4624*x^6 + 5077*x^5 + 5763*x^4 + 5464*x^3 + 1060*x^2 - 224*x + 2416)
 
gp: K = bnfinit(x^16 + 14*x^14 - 19*x^13 + 20*x^12 - 192*x^11 - 88*x^10 - 339*x^9 + 706*x^8 + 1248*x^7 + 4624*x^6 + 5077*x^5 + 5763*x^4 + 5464*x^3 + 1060*x^2 - 224*x + 2416, 1)
 

Normalized defining polynomial

\( x^{16} + 14 x^{14} - 19 x^{13} + 20 x^{12} - 192 x^{11} - 88 x^{10} - 339 x^{9} + 706 x^{8} + 1248 x^{7} + 4624 x^{6} + 5077 x^{5} + 5763 x^{4} + 5464 x^{3} + 1060 x^{2} - 224 x + 2416 \)

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:  \(616308629868476806640625=3^{12}\cdot 5^{12}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.68$
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:  $3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{12} a^{9} + \frac{1}{6} a^{8} - \frac{1}{4} a^{7} + \frac{5}{12} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{5}{12} a^{3} - \frac{5}{24} a^{2} - \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{7}{24} a^{6} - \frac{1}{2} a^{5} + \frac{5}{24} a^{4} + \frac{3}{8} a^{3} - \frac{11}{24} a^{2} - \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{15312} a^{14} + \frac{139}{7656} a^{13} + \frac{13}{957} a^{12} + \frac{141}{5104} a^{11} + \frac{497}{7656} a^{10} + \frac{353}{1914} a^{9} + \frac{211}{1914} a^{8} + \frac{1071}{5104} a^{7} - \frac{409}{1914} a^{6} - \frac{281}{7656} a^{5} + \frac{43}{696} a^{4} + \frac{7037}{15312} a^{3} - \frac{3541}{15312} a^{2} - \frac{549}{2552} a - \frac{809}{3828}$, $\frac{1}{4683606409897401063264} a^{15} + \frac{20534543818220803}{780601068316233510544} a^{14} + \frac{15891029355787559351}{780601068316233510544} a^{13} - \frac{29201482984127064485}{1561202136632467021088} a^{12} - \frac{22850444110737805739}{2341803204948700531632} a^{11} + \frac{24649385376721971215}{292725400618587566454} a^{10} - \frac{57808420884329426957}{585450801237175132908} a^{9} + \frac{614583158165051631661}{4683606409897401063264} a^{8} + \frac{130616855427120974449}{1170901602474350265816} a^{7} + \frac{120255658612838123509}{585450801237175132908} a^{6} - \frac{19284551197636515842}{48787566769764594409} a^{5} + \frac{363342228997343243691}{1561202136632467021088} a^{4} + \frac{1531800520257116330681}{4683606409897401063264} a^{3} - \frac{56840381620213548535}{2341803204948700531632} a^{2} + \frac{5109702501704730667}{1170901602474350265816} a + \frac{32717873025936492883}{97575133539529188818}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 10253.3634365 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.9225.1, 4.4.5125.1, 8.0.31402130625.1, 8.0.785053265625.2, 8.8.2127515625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$