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Properties

Label 16.10.7358939853...6875.1
Degree $16$
Signature $[10, 3]$
Discriminant $-\,3^{8}\cdot 5^{14}\cdot 179^{5}$
Root discriminant $35.82$
Ramified primes $3, 5, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2371, -5631, 2491, -6414, 7738, 6619, -201, 2193, -3997, -2056, 1316, 348, -82, -2, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 11*x^14 - 2*x^13 - 82*x^12 + 348*x^11 + 1316*x^10 - 2056*x^9 - 3997*x^8 + 2193*x^7 - 201*x^6 + 6619*x^5 + 7738*x^4 - 6414*x^3 + 2491*x^2 - 5631*x + 2371)
 
gp: K = bnfinit(x^16 - 2*x^15 - 11*x^14 - 2*x^13 - 82*x^12 + 348*x^11 + 1316*x^10 - 2056*x^9 - 3997*x^8 + 2193*x^7 - 201*x^6 + 6619*x^5 + 7738*x^4 - 6414*x^3 + 2491*x^2 - 5631*x + 2371, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 11 x^{14} - 2 x^{13} - 82 x^{12} + 348 x^{11} + 1316 x^{10} - 2056 x^{9} - 3997 x^{8} + 2193 x^{7} - 201 x^{6} + 6619 x^{5} + 7738 x^{4} - 6414 x^{3} + 2491 x^{2} - 5631 x + 2371 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7358939853847283935546875=-\,3^{8}\cdot 5^{14}\cdot 179^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.82$
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:  $3, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{53718257646519978232214709249543} a^{15} - \frac{232035879819051972016438244534}{53718257646519978232214709249543} a^{14} - \frac{4310497601383765002337926762794}{17906085882173326077404903083181} a^{13} - \frac{23363378857411182776082569242192}{53718257646519978232214709249543} a^{12} - \frac{1703451864794968924239261937019}{17906085882173326077404903083181} a^{11} + \frac{17652519381280401770811179356217}{53718257646519978232214709249543} a^{10} + \frac{19683374477637969282139649444119}{53718257646519978232214709249543} a^{9} + \frac{8665925185949707611785398620713}{53718257646519978232214709249543} a^{8} - \frac{7683996516993324829736003098253}{53718257646519978232214709249543} a^{7} + \frac{8220393910440024682392918044285}{17906085882173326077404903083181} a^{6} - \frac{23354044466668117010390456618269}{53718257646519978232214709249543} a^{5} + \frac{2151508350989036268867352742554}{17906085882173326077404903083181} a^{4} + \frac{17962913020981275424589951339176}{53718257646519978232214709249543} a^{3} + \frac{3252086736341513171686684888521}{17906085882173326077404903083181} a^{2} + \frac{8435256271320405756451122733167}{17906085882173326077404903083181} a + \frac{6681703501452690724626861572453}{53718257646519978232214709249543}$

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:  $12$
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:  \( 4370225.78406 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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 $16$ R R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
179Data not computed