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Properties

Label 16.0.42819656572...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $71.12$
Ramified primes $2, 5, 761$
Class number $12792$ (GRH)
Class group $[2, 2, 3198]$ (GRH)
Galois group 16T1163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5722919, -10467112, 14677804, -11014656, 8382885, -3941164, 2478520, -703088, 417537, -66432, 41698, -3192, 2453, -60, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 78*x^14 - 60*x^13 + 2453*x^12 - 3192*x^11 + 41698*x^10 - 66432*x^9 + 417537*x^8 - 703088*x^7 + 2478520*x^6 - 3941164*x^5 + 8382885*x^4 - 11014656*x^3 + 14677804*x^2 - 10467112*x + 5722919)
 
gp: K = bnfinit(x^16 + 78*x^14 - 60*x^13 + 2453*x^12 - 3192*x^11 + 41698*x^10 - 66432*x^9 + 417537*x^8 - 703088*x^7 + 2478520*x^6 - 3941164*x^5 + 8382885*x^4 - 11014656*x^3 + 14677804*x^2 - 10467112*x + 5722919, 1)
 

Normalized defining polynomial

\( x^{16} + 78 x^{14} - 60 x^{13} + 2453 x^{12} - 3192 x^{11} + 41698 x^{10} - 66432 x^{9} + 417537 x^{8} - 703088 x^{7} + 2478520 x^{6} - 3941164 x^{5} + 8382885 x^{4} - 11014656 x^{3} + 14677804 x^{2} - 10467112 x + 5722919 \)

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:  \(428196565723310037401600000000=2^{32}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.12$
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:  $2, 5, 761$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2642953} a^{14} + \frac{1318125}{2642953} a^{13} - \frac{399065}{2642953} a^{12} + \frac{532319}{2642953} a^{11} + \frac{558097}{2642953} a^{10} - \frac{765719}{2642953} a^{9} + \frac{1029580}{2642953} a^{8} + \frac{175198}{2642953} a^{7} + \frac{1029876}{2642953} a^{6} - \frac{148167}{2642953} a^{5} + \frac{59462}{2642953} a^{4} - \frac{962742}{2642953} a^{3} - \frac{203824}{2642953} a^{2} - \frac{347380}{2642953} a + \frac{116070}{2642953}$, $\frac{1}{1564350389543534173532721394356042632553} a^{15} + \frac{290153777650913450622349689363581}{1564350389543534173532721394356042632553} a^{14} - \frac{108172867152996490426483338135403570613}{1564350389543534173532721394356042632553} a^{13} - \frac{403122628627347856463143252519183782391}{1564350389543534173532721394356042632553} a^{12} + \frac{38891975181269966622462125696256163502}{521450129847844724510907131452014210851} a^{11} + \frac{831182108054485504418063987411968689}{521450129847844724510907131452014210851} a^{10} - \frac{92699466627649063160317786979939350487}{1564350389543534173532721394356042632553} a^{9} - \frac{22440695496911040925827218273407755352}{1564350389543534173532721394356042632553} a^{8} - \frac{382522272579193804210467307356843323429}{1564350389543534173532721394356042632553} a^{7} - \frac{83546249831587917939266934819502208380}{521450129847844724510907131452014210851} a^{6} + \frac{730235895645330167990867582438142369748}{1564350389543534173532721394356042632553} a^{5} + \frac{308985638172101517412319541618317657314}{1564350389543534173532721394356042632553} a^{4} - \frac{47625274031121935297865140092497064576}{1564350389543534173532721394356042632553} a^{3} - \frac{750565522638309181684759216165159797542}{1564350389543534173532721394356042632553} a^{2} + \frac{118175479238191227167077843835088127568}{521450129847844724510907131452014210851} a + \frac{32873040721240082611280663515325506190}{1564350389543534173532721394356042632553}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3198}$, which has order $12792$ (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:  \( 9562.41394678 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1163 are not computed
Character table for t16n1163 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
761Data not computed