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Properties

Label 16.0.64790048254...0625.7
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{6}$
Root discriminant $23.08$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52081, -22120, 98444, -26025, 85157, -20250, 43177, -8500, 13365, -1875, 2623, -215, 332, -10, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 26*x^14 - 10*x^13 + 332*x^12 - 215*x^11 + 2623*x^10 - 1875*x^9 + 13365*x^8 - 8500*x^7 + 43177*x^6 - 20250*x^5 + 85157*x^4 - 26025*x^3 + 98444*x^2 - 22120*x + 52081)
 
gp: K = bnfinit(x^16 + 26*x^14 - 10*x^13 + 332*x^12 - 215*x^11 + 2623*x^10 - 1875*x^9 + 13365*x^8 - 8500*x^7 + 43177*x^6 - 20250*x^5 + 85157*x^4 - 26025*x^3 + 98444*x^2 - 22120*x + 52081, 1)
 

Normalized defining polynomial

\( x^{16} + 26 x^{14} - 10 x^{13} + 332 x^{12} - 215 x^{11} + 2623 x^{10} - 1875 x^{9} + 13365 x^{8} - 8500 x^{7} + 43177 x^{6} - 20250 x^{5} + 85157 x^{4} - 26025 x^{3} + 98444 x^{2} - 22120 x + 52081 \)

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:  \(6479004825445556640625=5^{14}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.08$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{155381049276737669238438328300381} a^{15} + \frac{33634143029712074690174680522163}{155381049276737669238438328300381} a^{14} + \frac{44360985441114959941306387334454}{155381049276737669238438328300381} a^{13} + \frac{29806659855362248204835678530282}{155381049276737669238438328300381} a^{12} - \frac{53659155497077818752470404192591}{155381049276737669238438328300381} a^{11} + \frac{38368584523298600728205760656952}{155381049276737669238438328300381} a^{10} + \frac{66203388255976604748455826314479}{155381049276737669238438328300381} a^{9} + \frac{74504192217998418569924418556997}{155381049276737669238438328300381} a^{8} + \frac{49635370696143061844234627794603}{155381049276737669238438328300381} a^{7} + \frac{1036616606722803598610423347945}{14125549934248879021676211663671} a^{6} - \frac{61400185004637217098277035301874}{155381049276737669238438328300381} a^{5} - \frac{24225862171514828862439109519067}{155381049276737669238438328300381} a^{4} + \frac{30839303957604108144288695224817}{155381049276737669238438328300381} a^{3} + \frac{29786541349128758064039766781777}{155381049276737669238438328300381} a^{2} + \frac{71226678088368695525156411717540}{155381049276737669238438328300381} a - \frac{7573944287642495751299324161656}{155381049276737669238438328300381}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{1446013908848045961903316}{58612240391074186811934488231} a^{15} + \frac{13498173462376888609878132}{58612240391074186811934488231} a^{14} + \frac{44231237891704419064065838}{58612240391074186811934488231} a^{13} + \frac{294939422384315265717958700}{58612240391074186811934488231} a^{12} + \frac{468323813340483497914084337}{58612240391074186811934488231} a^{11} + \frac{3144981044637501827099392161}{58612240391074186811934488231} a^{10} + \frac{2454277546888867290765619263}{58612240391074186811934488231} a^{9} + \frac{20761657525587706782258253659}{58612240391074186811934488231} a^{8} + \frac{7891502494976662346951145495}{58612240391074186811934488231} a^{7} + \frac{87279958555630573384754367649}{58612240391074186811934488231} a^{6} + \frac{20090746692531297995625315629}{58612240391074186811934488231} a^{5} + \frac{219757093384197499808099105655}{58612240391074186811934488231} a^{4} + \frac{54238516158720357475288052520}{58612240391074186811934488231} a^{3} + \frac{294203616202598122223943062424}{58612240391074186811934488231} a^{2} + \frac{60288205278177067391681977966}{58612240391074186811934488231} a + \frac{170268070813244555363248933253}{58612240391074186811934488231} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 36899.0417494 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.1578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$