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Properties

Label 16.0.95908900025...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{7}$
Root discriminant $27.31$
Ramified primes $5, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114211, -2951, -83455, -57600, 91200, 34507, -52137, -8660, 15780, 1080, -2757, -128, 325, 15, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 25*x^14 + 15*x^13 + 325*x^12 - 128*x^11 - 2757*x^10 + 1080*x^9 + 15780*x^8 - 8660*x^7 - 52137*x^6 + 34507*x^5 + 91200*x^4 - 57600*x^3 - 83455*x^2 - 2951*x + 114211)
 
gp: K = bnfinit(x^16 - x^15 - 25*x^14 + 15*x^13 + 325*x^12 - 128*x^11 - 2757*x^10 + 1080*x^9 + 15780*x^8 - 8660*x^7 - 52137*x^6 + 34507*x^5 + 91200*x^4 - 57600*x^3 - 83455*x^2 - 2951*x + 114211, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 25 x^{14} + 15 x^{13} + 325 x^{12} - 128 x^{11} - 2757 x^{10} + 1080 x^{9} + 15780 x^{8} - 8660 x^{7} - 52137 x^{6} + 34507 x^{5} + 91200 x^{4} - 57600 x^{3} - 83455 x^{2} - 2951 x + 114211 \)

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:  \(95908900025054931640625=5^{15}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.31$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{429001316691763925014863520215669993419} a^{15} - \frac{62901574913578909106817215115430528673}{429001316691763925014863520215669993419} a^{14} - \frac{138294086616511108249365915012365013043}{429001316691763925014863520215669993419} a^{13} - \frac{13785720357676405900712536894281354124}{429001316691763925014863520215669993419} a^{12} + \frac{90151533814632051884275413182648957976}{429001316691763925014863520215669993419} a^{11} + \frac{166014982338922755125965397965865190533}{429001316691763925014863520215669993419} a^{10} + \frac{30966791363942865032751594336422881952}{429001316691763925014863520215669993419} a^{9} - \frac{149200208456503349237178044211945251575}{429001316691763925014863520215669993419} a^{8} + \frac{72846676671345781547261573230515178082}{429001316691763925014863520215669993419} a^{7} - \frac{157740514407453755897415673561841046866}{429001316691763925014863520215669993419} a^{6} + \frac{120736811446463143779581153299569965216}{429001316691763925014863520215669993419} a^{5} + \frac{81480354181244750084016790646941260352}{429001316691763925014863520215669993419} a^{4} - \frac{1716593018540463391385549627017204898}{10463446748579607927191793175991951059} a^{3} - \frac{78631072563784841687984733939231961672}{429001316691763925014863520215669993419} a^{2} + \frac{100579767795593127896988907953099672683}{429001316691763925014863520215669993419} a + \frac{78040508472082756324482187487341176}{2370173020396485773562781879644585599}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{648993816780874903190742429864}{23467059607885997758047345343015699} a^{15} + \frac{493691776511630340493188581220}{23467059607885997758047345343015699} a^{14} - \frac{15039804056742810028474520282041}{23467059607885997758047345343015699} a^{13} - \frac{17884219030350051560764550691012}{23467059607885997758047345343015699} a^{12} + \frac{173550601782775447170477380766785}{23467059607885997758047345343015699} a^{11} + \frac{245096036445770652941838187979105}{23467059607885997758047345343015699} a^{10} - \frac{1294643729658766015863479421236735}{23467059607885997758047345343015699} a^{9} - \frac{1842117215831005061311786690336554}{23467059607885997758047345343015699} a^{8} + \frac{6582017582288722678779936994141020}{23467059607885997758047345343015699} a^{7} + \frac{8109363306686047242436557248935976}{23467059607885997758047345343015699} a^{6} - \frac{18292679465844160830042192553121976}{23467059607885997758047345343015699} a^{5} - \frac{21839031992973350281943415928142264}{23467059607885997758047345343015699} a^{4} + \frac{623692187116800899728888737981251}{572367307509414579464569398610139} a^{3} + \frac{37987830732351311071960873695598458}{23467059607885997758047345343015699} a^{2} - \frac{16626480857684567426209195033657670}{23467059607885997758047345343015699} a - \frac{44480167561625722863125637622809136}{23467059607885997758047345343015699} \) (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:  \( 62915.747768 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$