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Properties

Label 16.2.21303603142...5827.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,43^{7}\cdot 97^{8}$
Root discriminant $51.05$
Ramified primes $43, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13824, 2916, -28458, -4171, 15269, 8672, -3584, -3960, -736, 1182, 450, -200, -80, 28, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 28*x^13 - 80*x^12 - 200*x^11 + 450*x^10 + 1182*x^9 - 736*x^8 - 3960*x^7 - 3584*x^6 + 8672*x^5 + 15269*x^4 - 4171*x^3 - 28458*x^2 + 2916*x + 13824)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 + 28*x^13 - 80*x^12 - 200*x^11 + 450*x^10 + 1182*x^9 - 736*x^8 - 3960*x^7 - 3584*x^6 + 8672*x^5 + 15269*x^4 - 4171*x^3 - 28458*x^2 + 2916*x + 13824, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} + 28 x^{13} - 80 x^{12} - 200 x^{11} + 450 x^{10} + 1182 x^{9} - 736 x^{8} - 3960 x^{7} - 3584 x^{6} + 8672 x^{5} + 15269 x^{4} - 4171 x^{3} - 28458 x^{2} + 2916 x + 13824 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2130360314266888344019505827=-\,43^{7}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.05$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{7}{24} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{144} a^{12} + \frac{1}{72} a^{11} - \frac{1}{72} a^{10} + \frac{5}{144} a^{9} + \frac{1}{24} a^{8} + \frac{3}{16} a^{6} - \frac{1}{24} a^{5} - \frac{5}{72} a^{4} - \frac{65}{144} a^{3} - \frac{17}{72} a^{2} + \frac{7}{36} a + \frac{1}{3}$, $\frac{1}{144} a^{13} - \frac{1}{48} a^{10} + \frac{1}{72} a^{9} - \frac{1}{12} a^{8} - \frac{1}{16} a^{7} + \frac{1}{12} a^{6} - \frac{1}{9} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{2448} a^{14} + \frac{1}{1224} a^{13} - \frac{1}{612} a^{12} - \frac{41}{2448} a^{11} + \frac{1}{72} a^{10} - \frac{11}{306} a^{9} + \frac{77}{816} a^{8} - \frac{19}{408} a^{7} - \frac{13}{153} a^{6} - \frac{575}{2448} a^{5} - \frac{61}{1224} a^{4} + \frac{23}{612} a^{3} + \frac{151}{306} a^{2} - \frac{74}{153} a + \frac{1}{51}$, $\frac{1}{1260537381481648608} a^{15} + \frac{1013955105287}{16160735660021136} a^{14} + \frac{492363625876639}{157567172685206076} a^{13} - \frac{743134823568629}{315134345370412152} a^{12} + \frac{2805428325585937}{315134345370412152} a^{11} + \frac{2103650930201833}{315134345370412152} a^{10} + \frac{8047239049875085}{210089563580274768} a^{9} + \frac{2690408367547}{237657877353252} a^{8} + \frac{72844809229771187}{315134345370412152} a^{7} - \frac{7894923349126457}{105044781790137384} a^{6} + \frac{2001428872009843}{315134345370412152} a^{5} - \frac{2085325632788117}{18537314433553656} a^{4} - \frac{506779629590421199}{1260537381481648608} a^{3} - \frac{156372689267633453}{630268690740824304} a^{2} - \frac{44715292388344319}{105044781790137384} a + \frac{1401448549200463}{4376865907922391}$

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:  $8$
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:  \( 1122360429.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.2.404587.1, 8.2.7038697544467.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
43Data not computed
$97$97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$