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Properties

Label 16.16.3034137770...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 5^{14}\cdot 41^{4}$
Root discriminant $69.60$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1112399, 1846232, 6583356, -8237872, -3144804, 5810256, 137564, -1675736, 158082, 238504, -33076, -17344, 2556, 608, -84, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 84*x^14 + 608*x^13 + 2556*x^12 - 17344*x^11 - 33076*x^10 + 238504*x^9 + 158082*x^8 - 1675736*x^7 + 137564*x^6 + 5810256*x^5 - 3144804*x^4 - 8237872*x^3 + 6583356*x^2 + 1846232*x - 1112399)
 
gp: K = bnfinit(x^16 - 8*x^15 - 84*x^14 + 608*x^13 + 2556*x^12 - 17344*x^11 - 33076*x^10 + 238504*x^9 + 158082*x^8 - 1675736*x^7 + 137564*x^6 + 5810256*x^5 - 3144804*x^4 - 8237872*x^3 + 6583356*x^2 + 1846232*x - 1112399, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 84 x^{14} + 608 x^{13} + 2556 x^{12} - 17344 x^{11} - 33076 x^{10} + 238504 x^{9} + 158082 x^{8} - 1675736 x^{7} + 137564 x^{6} + 5810256 x^{5} - 3144804 x^{4} - 8237872 x^{3} + 6583356 x^{2} + 1846232 x - 1112399 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(303413777032806400000000000000=2^{44}\cdot 5^{14}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.60$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{4}$, $\frac{1}{44} a^{13} + \frac{3}{44} a^{12} - \frac{3}{44} a^{10} + \frac{1}{11} a^{8} + \frac{4}{11} a^{7} + \frac{2}{11} a^{6} + \frac{5}{44} a^{5} - \frac{9}{44} a^{4} + \frac{1}{11} a^{3} + \frac{1}{44} a^{2} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{7348} a^{14} + \frac{41}{7348} a^{13} - \frac{733}{7348} a^{12} - \frac{333}{7348} a^{11} + \frac{359}{7348} a^{10} - \frac{73}{7348} a^{9} - \frac{899}{7348} a^{8} + \frac{20}{167} a^{7} + \frac{1409}{7348} a^{6} + \frac{3481}{7348} a^{5} - \frac{129}{7348} a^{4} - \frac{2685}{7348} a^{3} + \frac{3375}{7348} a^{2} - \frac{3573}{7348} a + \frac{2649}{7348}$, $\frac{1}{1145433917237061268379862235257508} a^{15} + \frac{16938798161449891551871910925}{286358479309265317094965558814377} a^{14} - \frac{12159153567929080818977675075479}{1145433917237061268379862235257508} a^{13} + \frac{54808605956391274071372818855023}{572716958618530634189931117628754} a^{12} + \frac{8829176762315254693345881472125}{286358479309265317094965558814377} a^{11} + \frac{80886645886834908457108906015}{26032589028115028826815050801307} a^{10} - \frac{59267923810149301609165038007459}{572716958618530634189931117628754} a^{9} - \frac{115703794797297496720966715201257}{1145433917237061268379862235257508} a^{8} - \frac{49755972836904218085968061315169}{104130356112460115307260203205228} a^{7} - \frac{130506678348764556059403237195744}{286358479309265317094965558814377} a^{6} + \frac{57679524346887964859443508913229}{1145433917237061268379862235257508} a^{5} - \frac{86877152031206215400656803511359}{572716958618530634189931117628754} a^{4} + \frac{8455113538951240877055558463479}{286358479309265317094965558814377} a^{3} + \frac{39218901928412251158696056526991}{286358479309265317094965558814377} a^{2} - \frac{200551016299876875528045613342377}{572716958618530634189931117628754} a - \frac{185834647812819262512263144385229}{1145433917237061268379862235257508}$

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:  $15$
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:  \( 4017204622.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{20})^+\), 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
41Data not computed