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Properties

Label 16.12.6690022700...9293.2
Degree $16$
Signature $[12, 2]$
Discriminant $3^{8}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $267.05$
Ramified primes $3, 13, 1327$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11794897, -7770217, -23034358, 26735463, -4438837, -8440364, 5070510, 936994, -834951, -58366, 35406, 2329, 943, -49, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 78*x^14 - 49*x^13 + 943*x^12 + 2329*x^11 + 35406*x^10 - 58366*x^9 - 834951*x^8 + 936994*x^7 + 5070510*x^6 - 8440364*x^5 - 4438837*x^4 + 26735463*x^3 - 23034358*x^2 - 7770217*x + 11794897)
 
gp: K = bnfinit(x^16 - 78*x^14 - 49*x^13 + 943*x^12 + 2329*x^11 + 35406*x^10 - 58366*x^9 - 834951*x^8 + 936994*x^7 + 5070510*x^6 - 8440364*x^5 - 4438837*x^4 + 26735463*x^3 - 23034358*x^2 - 7770217*x + 11794897, 1)
 

Normalized defining polynomial

\( x^{16} - 78 x^{14} - 49 x^{13} + 943 x^{12} + 2329 x^{11} + 35406 x^{10} - 58366 x^{9} - 834951 x^{8} + 936994 x^{7} + 5070510 x^{6} - 8440364 x^{5} - 4438837 x^{4} + 26735463 x^{3} - 23034358 x^{2} - 7770217 x + 11794897 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(669002270083080728394108412888785769293=3^{8}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $267.05$
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:  $3, 13, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2541787651741253761710781342066933923536979352930717918} a^{15} + \frac{603776162978454859195371009024635127576418294114455781}{2541787651741253761710781342066933923536979352930717918} a^{14} - \frac{171964802514411508528311340707750242533591067288113635}{1270893825870626880855390671033466961768489676465358959} a^{13} + \frac{350055754919141729936826359342541565358850408702884295}{2541787651741253761710781342066933923536979352930717918} a^{12} - \frac{95883021236013168785479185543726128513609012562141742}{1270893825870626880855390671033466961768489676465358959} a^{11} - \frac{283603270506272092639049598066509933796284075140600319}{2541787651741253761710781342066933923536979352930717918} a^{10} + \frac{617234754016909934601021094376769408382572702836935907}{1270893825870626880855390671033466961768489676465358959} a^{9} + \frac{48912321671073355844103599182854367737729840341788423}{1270893825870626880855390671033466961768489676465358959} a^{8} + \frac{1874311744549097078734957621437288485287244483590077}{1270893825870626880855390671033466961768489676465358959} a^{7} + \frac{1233982002184648502176434156640276697042682186407316209}{2541787651741253761710781342066933923536979352930717918} a^{6} - \frac{349547085317072701010283328222615429968961781897073091}{2541787651741253761710781342066933923536979352930717918} a^{5} + \frac{557827375437586601760196106950486624010368878579906620}{1270893825870626880855390671033466961768489676465358959} a^{4} - \frac{973538344827556047631856652713157365087758790176930759}{2541787651741253761710781342066933923536979352930717918} a^{3} + \frac{151734346339453880505675969842658038166553404588399396}{1270893825870626880855390671033466961768489676465358959} a^{2} + \frac{275010581915927790010379536659718969374494949071047358}{1270893825870626880855390671033466961768489676465358959} a - \frac{453726987403095916283849215663257908938053088245272122}{1270893825870626880855390671033466961768489676465358959}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  $13$
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:  \( 1324767734840 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1327Data not computed