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Properties

Label 16.0.21078441977...0000.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{8}\cdot 61^{8}$
Root discriminant $139.71$
Ramified primes $2, 5, 61$
Class number $3631680$ (GRH)
Class group $[6, 605280]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![123000632449, -42254476064, 48544576244, -13143868456, 8014040880, -1823888920, 767003836, -143269808, 44663097, -6146760, 1482352, -142264, 27290, -1680, 260, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 260*x^14 - 1680*x^13 + 27290*x^12 - 142264*x^11 + 1482352*x^10 - 6146760*x^9 + 44663097*x^8 - 143269808*x^7 + 767003836*x^6 - 1823888920*x^5 + 8014040880*x^4 - 13143868456*x^3 + 48544576244*x^2 - 42254476064*x + 123000632449)
 
gp: K = bnfinit(x^16 - 8*x^15 + 260*x^14 - 1680*x^13 + 27290*x^12 - 142264*x^11 + 1482352*x^10 - 6146760*x^9 + 44663097*x^8 - 143269808*x^7 + 767003836*x^6 - 1823888920*x^5 + 8014040880*x^4 - 13143868456*x^3 + 48544576244*x^2 - 42254476064*x + 123000632449, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 260 x^{14} - 1680 x^{13} + 27290 x^{12} - 142264 x^{11} + 1482352 x^{10} - 6146760 x^{9} + 44663097 x^{8} - 143269808 x^{7} + 767003836 x^{6} - 1823888920 x^{5} + 8014040880 x^{4} - 13143868456 x^{3} + 48544576244 x^{2} - 42254476064 x + 123000632449 \)

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:  \(21078441977020355377207705600000000=2^{48}\cdot 5^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.71$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(1219,·)$, $\chi_{4880}(1221,·)$, $\chi_{4880}(2439,·)$, $\chi_{4880}(2441,·)$, $\chi_{4880}(3659,·)$, $\chi_{4880}(3661,·)$, $\chi_{4880}(4879,·)$, $\chi_{4880}(731,·)$, $\chi_{4880}(1951,·)$, $\chi_{4880}(3171,·)$, $\chi_{4880}(4391,·)$, $\chi_{4880}(489,·)$, $\chi_{4880}(1709,·)$, $\chi_{4880}(2929,·)$, $\chi_{4880}(4149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{15841} a^{8} - \frac{4}{15841} a^{7} + \frac{1036}{2263} a^{6} - \frac{843}{2263} a^{5} + \frac{437}{2263} a^{4} - \frac{224}{2263} a^{3} - \frac{5193}{15841} a^{2} + \frac{2354}{15841} a - \frac{2054}{15841}$, $\frac{1}{15841} a^{9} + \frac{447}{15841} a^{7} + \frac{1038}{2263} a^{6} - \frac{672}{2263} a^{5} - \frac{739}{2263} a^{4} + \frac{4376}{15841} a^{3} - \frac{2577}{15841} a^{2} + \frac{573}{15841} a - \frac{1427}{15841}$, $\frac{1}{15841} a^{10} + \frac{2}{15841} a^{7} + \frac{151}{2263} a^{6} + \frac{424}{2263} a^{5} - \frac{671}{15841} a^{4} + \frac{1315}{15841} a^{3} - \frac{969}{2263} a^{2} - \frac{1370}{15841} a - \frac{7429}{15841}$, $\frac{1}{110887} a^{11} - \frac{2}{110887} a^{10} + \frac{3}{110887} a^{9} - \frac{2}{110887} a^{8} + \frac{4673}{110887} a^{7} - \frac{908}{15841} a^{6} + \frac{50408}{110887} a^{5} + \frac{22425}{110887} a^{4} - \frac{5854}{110887} a^{3} + \frac{9396}{110887} a^{2} + \frac{5718}{110887} a + \frac{16530}{110887}$, $\frac{1}{110887} a^{12} - \frac{1}{110887} a^{10} - \frac{3}{110887} a^{9} + \frac{2696}{110887} a^{7} - \frac{52219}{110887} a^{6} - \frac{13812}{110887} a^{5} - \frac{13728}{110887} a^{4} - \frac{30494}{110887} a^{3} + \frac{4413}{110887} a^{2} - \frac{4899}{110887} a + \frac{1847}{110887}$, $\frac{1}{110887} a^{13} + \frac{2}{110887} a^{10} + \frac{3}{110887} a^{9} - \frac{1}{110887} a^{8} - \frac{5070}{110887} a^{7} - \frac{40797}{110887} a^{6} - \frac{1011}{15841} a^{5} - \frac{51133}{110887} a^{4} + \frac{19818}{110887} a^{3} - \frac{19611}{110887} a^{2} + \frac{6186}{110887} a + \frac{34912}{110887}$, $\frac{1}{11934331935659724353} a^{14} - \frac{1}{1704904562237103479} a^{13} - \frac{2130490067753}{11934331935659724353} a^{12} + \frac{751937670977}{702019525627042609} a^{11} + \frac{203853142591393}{11934331935659724353} a^{10} + \frac{3111952118915}{100288503661006087} a^{9} + \frac{102818519980356}{11934331935659724353} a^{8} + \frac{680071935334021749}{11934331935659724353} a^{7} + \frac{1997296413035854023}{11934331935659724353} a^{6} - \frac{590249197803974805}{1704904562237103479} a^{5} - \frac{4498930224391006566}{11934331935659724353} a^{4} + \frac{3041880678800884717}{11934331935659724353} a^{3} - \frac{109976108465414952}{518883997202596711} a^{2} + \frac{5755102308624122718}{11934331935659724353} a + \frac{41535334303538977}{151067492856452207}$, $\frac{1}{20942950462960531621137697} a^{15} + \frac{877417}{20942950462960531621137697} a^{14} - \frac{58141326730804213825}{20942950462960531621137697} a^{13} - \frac{73292185226443663}{910563063606979635701639} a^{12} + \frac{3511527514401156437}{20942950462960531621137697} a^{11} + \frac{408088335173145136274}{20942950462960531621137697} a^{10} + \frac{607247918002784587737}{20942950462960531621137697} a^{9} + \frac{31090823510744969444}{20942950462960531621137697} a^{8} + \frac{674796702457989593218930}{20942950462960531621137697} a^{7} - \frac{2753856990864004876174208}{20942950462960531621137697} a^{6} - \frac{4154667257069337424778209}{20942950462960531621137697} a^{5} + \frac{6830944621220336931566592}{20942950462960531621137697} a^{4} - \frac{2342673817966457977074729}{20942950462960531621137697} a^{3} - \frac{2351690751413777036621709}{20942950462960531621137697} a^{2} + \frac{278065137070728514652906}{675579047192275213585087} a + \frac{68969291854400690899200}{265100638771652299001743}$

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

Class group and class number

$C_{6}\times C_{605280}$, which has order $3631680$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-122}) \), \(\Q(\sqrt{-305}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-61}) \), \(\Q(\sqrt{-610}) \), \(\Q(\sqrt{10}, \sqrt{-122})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-61})\), \(\Q(\sqrt{2}, \sqrt{-61})\), \(\Q(\sqrt{5}, \sqrt{-122})\), \(\Q(\sqrt{2}, \sqrt{-305})\), \(\Q(\sqrt{5}, \sqrt{-61})\), 4.0.7620608.2, 4.0.190515200.2, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.0.567125647360000.45, 8.0.36296041431040000.36, 8.8.2621440000.1, 8.0.232294665158656.11, 8.0.145184165724160000.6, 8.0.145184165724160000.5, 8.0.145184165724160000.10

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
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}$
$61$61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$