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Properties

Label 16.0.37780199833...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $16.73$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -256, 128, -576, 608, -16, 360, -348, 25, -174, 90, -2, 38, -18, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 18*x^13 + 38*x^12 - 2*x^11 + 90*x^10 - 174*x^9 + 25*x^8 - 348*x^7 + 360*x^6 - 16*x^5 + 608*x^4 - 576*x^3 + 128*x^2 - 256*x + 256)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 18*x^13 + 38*x^12 - 2*x^11 + 90*x^10 - 174*x^9 + 25*x^8 - 348*x^7 + 360*x^6 - 16*x^5 + 608*x^4 - 576*x^3 + 128*x^2 - 256*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 18 x^{13} + 38 x^{12} - 2 x^{11} + 90 x^{10} - 174 x^{9} + 25 x^{8} - 348 x^{7} + 360 x^{6} - 16 x^{5} + 608 x^{4} - 576 x^{3} + 128 x^{2} - 256 x + 256 \)

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:  \(37780199833600000000=2^{24}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.73$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{480} a^{12} + \frac{1}{240} a^{11} - \frac{19}{240} a^{10} + \frac{17}{80} a^{9} + \frac{31}{240} a^{8} - \frac{7}{80} a^{7} - \frac{23}{240} a^{6} - \frac{9}{80} a^{5} + \frac{113}{480} a^{4} - \frac{1}{20} a^{3} - \frac{1}{60} a^{2} - \frac{11}{30} a - \frac{11}{30}$, $\frac{1}{960} a^{13} - \frac{7}{160} a^{11} - \frac{31}{480} a^{10} + \frac{49}{480} a^{9} - \frac{83}{480} a^{8} + \frac{19}{480} a^{7} - \frac{101}{480} a^{6} - \frac{19}{960} a^{5} + \frac{23}{96} a^{4} - \frac{11}{24} a^{3} + \frac{1}{3} a^{2} - \frac{19}{60} a + \frac{11}{30}$, $\frac{1}{1084800} a^{14} - \frac{5}{21696} a^{13} - \frac{1}{7232} a^{12} + \frac{4151}{542400} a^{11} + \frac{29531}{542400} a^{10} - \frac{42747}{180800} a^{9} - \frac{241}{1600} a^{8} - \frac{25501}{180800} a^{7} - \frac{189}{3200} a^{6} + \frac{4341}{90400} a^{5} + \frac{5443}{33900} a^{4} + \frac{30337}{67800} a^{3} - \frac{117}{904} a^{2} - \frac{1067}{3390} a + \frac{1696}{8475}$, $\frac{1}{67257600} a^{15} + \frac{1}{11209600} a^{14} - \frac{1669}{6725760} a^{13} - \frac{2003}{11209600} a^{12} + \frac{309069}{11209600} a^{11} - \frac{167939}{2241920} a^{10} - \frac{3427}{14464} a^{9} + \frac{5733053}{33628800} a^{8} + \frac{1010971}{22419200} a^{7} - \frac{1541}{542400} a^{6} + \frac{195917}{16814400} a^{5} - \frac{2305639}{8407200} a^{4} + \frac{492727}{4203600} a^{3} + \frac{16813}{105090} a^{2} + \frac{160409}{350300} a + \frac{251681}{525450}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{6547}{1345152} a^{15} - \frac{81}{448384} a^{14} - \frac{133}{168144} a^{13} - \frac{25219}{336288} a^{12} + \frac{8857}{336288} a^{11} + \frac{68869}{336288} a^{10} + \frac{1567}{2712} a^{9} + \frac{12809}{112096} a^{8} - \frac{613415}{1345152} a^{7} - \frac{25205}{14464} a^{6} - \frac{148341}{112096} a^{5} + \frac{11941}{112096} a^{4} + \frac{9590}{3503} a^{3} + \frac{88603}{84072} a^{2} - \frac{2125}{3503} a - \frac{25795}{21018} \) (order $4$)
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:  \( 4301.25670415 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

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 $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(i, \sqrt{35})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(i, \sqrt{5})\), 4.2.19600.1 x2, 4.2.400.1 x2, 4.0.320.1 x2, 4.0.15680.1 x2, 8.0.384160000.1, 8.4.6146560000.2 x2, 8.0.6146560000.6 x2, 8.0.245862400.2 x2, 8.0.384160000.2 x2, 8.0.2560000.1, 8.0.6146560000.4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$