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Properties

Label 16.16.1010754082...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}$
Root discriminant $365.42$
Ramified primes $2, 3, 5, 13, 1249, 1511$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3640607603231, 1473945897104, -1673580561096, -528960304284, 221710558722, 52722098664, -14780454956, -2261845988, 542450360, 46124992, -10768336, -451356, 110802, 2088, -548, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 548*x^14 + 2088*x^13 + 110802*x^12 - 451356*x^11 - 10768336*x^10 + 46124992*x^9 + 542450360*x^8 - 2261845988*x^7 - 14780454956*x^6 + 52722098664*x^5 + 221710558722*x^4 - 528960304284*x^3 - 1673580561096*x^2 + 1473945897104*x + 3640607603231)
 
gp: K = bnfinit(x^16 - 4*x^15 - 548*x^14 + 2088*x^13 + 110802*x^12 - 451356*x^11 - 10768336*x^10 + 46124992*x^9 + 542450360*x^8 - 2261845988*x^7 - 14780454956*x^6 + 52722098664*x^5 + 221710558722*x^4 - 528960304284*x^3 - 1673580561096*x^2 + 1473945897104*x + 3640607603231, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 548 x^{14} + 2088 x^{13} + 110802 x^{12} - 451356 x^{11} - 10768336 x^{10} + 46124992 x^{9} + 542450360 x^{8} - 2261845988 x^{7} - 14780454956 x^{6} + 52722098664 x^{5} + 221710558722 x^{4} - 528960304284 x^{3} - 1673580561096 x^{2} + 1473945897104 x + 3640607603231 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(101075408216203634510751650650521600000000=2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $365.42$
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:  $2, 3, 5, 13, 1249, 1511$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{12}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{4} a^{9} + \frac{1}{6} a^{8} - \frac{1}{4} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{12} a^{14} - \frac{1}{4} a^{10} + \frac{1}{6} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{15} + \frac{1635405789823039219689084153160534393048002976176160002216949453296772150}{50471675212181373068939976635866099992543939723189326941049605358073550239} a^{14} + \frac{3597190930161803721770002659743141050919904766889728298006159004368849009}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{13} - \frac{1233425182887706069360480169127356046368697878579941043191013708565254561}{67295566949575164091919968847821466656725252964252435921399473810764733652} a^{12} - \frac{34761764598262021045381785862937732433658269507821085492717837287420420783}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{11} - \frac{24138970612934575529744902798844080742906012979864306135316736724579582371}{100943350424362746137879953271732199985087879446378653882099210716147100478} a^{10} + \frac{15769061198693656188391676084858571167093235870247827763693877573714410523}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{9} + \frac{667537592547314057311663693260014672647119080976415201432908440217773001}{67295566949575164091919968847821466656725252964252435921399473810764733652} a^{8} + \frac{17779900549436309916892945736054082561902685969768564544126304415361436459}{67295566949575164091919968847821466656725252964252435921399473810764733652} a^{7} - \frac{20973866088419269535918734897356259479145026480482268015454321313968480178}{50471675212181373068939976635866099992543939723189326941049605358073550239} a^{6} - \frac{1393243899778183486456490543806688850474097885444649138531962607248342925}{67295566949575164091919968847821466656725252964252435921399473810764733652} a^{5} - \frac{74551018457863463123163338916027155341644957547089140320525535156004922915}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{4} - \frac{13753473298969898604889713181703027769860939844715383579719972344638807995}{201886700848725492275759906543464399970175758892757307764198421432294200956} a^{3} - \frac{6612205962282376468883095044126484433332421744204290302995376634304797905}{33647783474787582045959984423910733328362626482126217960699736905382366826} a^{2} - \frac{81854996102410987793599819795523005766443415931596058768702611766518643925}{201886700848725492275759906543464399970175758892757307764198421432294200956} a + \frac{20072890744678144256185941633737320353480754207670927520720851318400675973}{67295566949575164091919968847821466656725252964252435921399473810764733652}$

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:  $15$
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:  \( 1082596963460000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.421149081600.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1249Data not computed
1511Data not computed