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Properties

Label 16.0.11445391768...321.35
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{12}$
Root discriminant $134.48$
Ramified primes $13, 53$
Class number $4096$ (GRH)
Class group $[2, 2, 4, 16, 16]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61062797, -64315489, 27985873, 935100, 8132377, 849452, 210951, 128673, 54723, 8403, 9142, -794, 795, -46, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 33*x^14 - 46*x^13 + 795*x^12 - 794*x^11 + 9142*x^10 + 8403*x^9 + 54723*x^8 + 128673*x^7 + 210951*x^6 + 849452*x^5 + 8132377*x^4 + 935100*x^3 + 27985873*x^2 - 64315489*x + 61062797)
 
gp: K = bnfinit(x^16 + 33*x^14 - 46*x^13 + 795*x^12 - 794*x^11 + 9142*x^10 + 8403*x^9 + 54723*x^8 + 128673*x^7 + 210951*x^6 + 849452*x^5 + 8132377*x^4 + 935100*x^3 + 27985873*x^2 - 64315489*x + 61062797, 1)
 

Normalized defining polynomial

\( x^{16} + 33 x^{14} - 46 x^{13} + 795 x^{12} - 794 x^{11} + 9142 x^{10} + 8403 x^{9} + 54723 x^{8} + 128673 x^{7} + 210951 x^{6} + 849452 x^{5} + 8132377 x^{4} + 935100 x^{3} + 27985873 x^{2} - 64315489 x + 61062797 \)

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:  \(11445391768549949624817238631584321=13^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.48$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{780} a^{12} - \frac{17}{260} a^{11} + \frac{8}{195} a^{10} - \frac{6}{65} a^{9} - \frac{2}{65} a^{8} - \frac{19}{52} a^{7} + \frac{101}{780} a^{6} - \frac{1}{10} a^{5} - \frac{89}{780} a^{4} - \frac{133}{780} a^{3} - \frac{21}{260} a^{2} - \frac{371}{780} a - \frac{181}{780}$, $\frac{1}{22620} a^{13} + \frac{1}{7540} a^{12} + \frac{1369}{11310} a^{11} - \frac{699}{3770} a^{10} + \frac{583}{3770} a^{9} - \frac{657}{7540} a^{8} - \frac{469}{22620} a^{7} + \frac{1871}{3770} a^{6} + \frac{2329}{22620} a^{5} - \frac{4939}{22620} a^{4} + \frac{401}{1508} a^{3} - \frac{4163}{22620} a^{2} - \frac{119}{348} a + \frac{648}{1885}$, $\frac{1}{22620} a^{14} + \frac{1}{7540} a^{12} + \frac{184}{1885} a^{11} - \frac{823}{5655} a^{10} + \frac{73}{580} a^{9} + \frac{752}{5655} a^{8} + \frac{3051}{7540} a^{7} + \frac{23}{52} a^{6} + \frac{2108}{5655} a^{5} + \frac{829}{5655} a^{4} - \frac{171}{377} a^{3} - \frac{1117}{5655} a^{2} + \frac{139}{1740} a - \frac{2471}{11310}$, $\frac{1}{2456660111500962946388652290023562325088714380} a^{15} - \frac{1225432840565185635440697331240184625029}{818886703833654315462884096674520775029571460} a^{14} - \frac{4989294082129533456949108409937074404491}{818886703833654315462884096674520775029571460} a^{13} - \frac{11040776121144102213693815048138444780096}{122833005575048147319432614501178116254435719} a^{12} + \frac{476606131719267429727697478550995624638896703}{2456660111500962946388652290023562325088714380} a^{11} - \frac{556461822043884132092896801790429101347482243}{2456660111500962946388652290023562325088714380} a^{10} - \frac{473755529690035110001585841445149075697474421}{2456660111500962946388652290023562325088714380} a^{9} - \frac{130763664502693247386237698759894909110997197}{818886703833654315462884096674520775029571460} a^{8} - \frac{1730582145234349908559095307959214548356743}{62991284910281101189452622821116982694582420} a^{7} - \frac{7502449734332508086749453488894809750445249}{204721675958413578865721024168630193757392865} a^{6} + \frac{58895260032370252914415731234645056309341556}{122833005575048147319432614501178116254435719} a^{5} - \frac{979069792719085121191512561254496576807504277}{2456660111500962946388652290023562325088714380} a^{4} - \frac{247651448809756405515539459432390902378331247}{818886703833654315462884096674520775029571460} a^{3} + \frac{485087636595983116777335947003096380139100389}{1228330055750481473194326145011781162544357190} a^{2} + \frac{142709093814763679656695258426814828095519299}{1228330055750481473194326145011781162544357190} a + \frac{189010025492816950760504488647269112137714317}{2456660111500962946388652290023562325088714380}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{16}\times C_{16}$, which has order $4096$ (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:  \( -1 \) (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:  \( 8979144.94 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.0.1935401.1 x2, 4.0.25160213.1 x2, 4.0.2197.1, 4.0.6171373.2, 8.0.633036318205369.6, 8.0.38085844705129.6, 8.8.106983137776707361.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$