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Properties

Label 16.0.68668573773...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 109^{12}$
Root discriminant $112.80$
Ramified primes $5, 109$
Class number $6800$ (GRH)
Class group $[20, 340]$ (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![4791131651, -398902253, 1801687213, -202336688, 311208069, -40610728, 33049817, -4384707, 2380153, -287877, 118502, -11714, 4015, -294, 89, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 89*x^14 - 294*x^13 + 4015*x^12 - 11714*x^11 + 118502*x^10 - 287877*x^9 + 2380153*x^8 - 4384707*x^7 + 33049817*x^6 - 40610728*x^5 + 311208069*x^4 - 202336688*x^3 + 1801687213*x^2 - 398902253*x + 4791131651)
 
gp: K = bnfinit(x^16 - 4*x^15 + 89*x^14 - 294*x^13 + 4015*x^12 - 11714*x^11 + 118502*x^10 - 287877*x^9 + 2380153*x^8 - 4384707*x^7 + 33049817*x^6 - 40610728*x^5 + 311208069*x^4 - 202336688*x^3 + 1801687213*x^2 - 398902253*x + 4791131651, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 89 x^{14} - 294 x^{13} + 4015 x^{12} - 11714 x^{11} + 118502 x^{10} - 287877 x^{9} + 2380153 x^{8} - 4384707 x^{7} + 33049817 x^{6} - 40610728 x^{5} + 311208069 x^{4} - 202336688 x^{3} + 1801687213 x^{2} - 398902253 x + 4791131651 \)

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:  \(686685737739964474054511962890625=5^{12}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.80$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4241780} a^{12} - \frac{3}{4241780} a^{11} - \frac{90829}{424178} a^{10} + \frac{90685}{424178} a^{9} + \frac{207349}{1060445} a^{8} - \frac{62165}{848356} a^{7} - \frac{1029569}{4241780} a^{6} + \frac{28058}{212089} a^{5} - \frac{1389}{4241780} a^{4} - \frac{156519}{848356} a^{3} + \frac{62005}{848356} a^{2} + \frac{462463}{4241780} a + \frac{966081}{4241780}$, $\frac{1}{21208900} a^{13} - \frac{1}{10604450} a^{12} - \frac{5150073}{21208900} a^{11} - \frac{212233}{2120890} a^{10} - \frac{96161}{5302225} a^{9} + \frac{2639461}{21208900} a^{8} - \frac{670197}{10604450} a^{7} + \frac{5894261}{21208900} a^{6} + \frac{6922441}{21208900} a^{5} + \frac{1924894}{5302225} a^{4} - \frac{47257}{2120890} a^{3} + \frac{1446689}{10604450} a^{2} - \frac{1763754}{5302225} a + \frac{7328751}{21208900}$, $\frac{1}{21208900} a^{14} - \frac{1}{10604450} a^{12} + \frac{3940649}{21208900} a^{11} + \frac{2553123}{10604450} a^{10} - \frac{858377}{21208900} a^{9} + \frac{1073032}{5302225} a^{8} + \frac{1173112}{5302225} a^{7} - \frac{1953053}{5302225} a^{6} - \frac{932873}{5302225} a^{5} - \frac{1732743}{21208900} a^{4} + \frac{1710663}{21208900} a^{3} + \frac{464763}{4241780} a^{2} - \frac{1174689}{5302225} a + \frac{8412577}{21208900}$, $\frac{1}{31210814372277206150865600586362504100} a^{15} + \frac{274750073631199162661286796641}{15605407186138603075432800293181252050} a^{14} - \frac{80684703437488727733637673431}{7802703593069301537716400146590626025} a^{13} - \frac{1812385047743600015266366823683}{15605407186138603075432800293181252050} a^{12} - \frac{386732337673000029442658978028217139}{6242162874455441230173120117272500820} a^{11} + \frac{1351215261422832877144391092624162411}{6242162874455441230173120117272500820} a^{10} + \frac{54116250632782567497564623681936441}{15605407186138603075432800293181252050} a^{9} + \frac{881497561246274050780127020623752558}{7802703593069301537716400146590626025} a^{8} - \frac{4026508495460901815242861353219621883}{31210814372277206150865600586362504100} a^{7} - \frac{12422387262117071681615097285002339613}{31210814372277206150865600586362504100} a^{6} + \frac{11136712414577778869935224874962460911}{31210814372277206150865600586362504100} a^{5} - \frac{1216678087497130889281718217060576683}{3121081437227720615086560058636250410} a^{4} + \frac{3125627986536143214901589056937195899}{7802703593069301537716400146590626025} a^{3} + \frac{621057485838523801281828413331013633}{31210814372277206150865600586362504100} a^{2} - \frac{341670204810944558934830918564643237}{7802703593069301537716400146590626025} a + \frac{780826456039261776521391171802067847}{31210814372277206150865600586362504100}$

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

Class group and class number

$C_{20}\times C_{340}$, which has order $6800$ (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:  \( 1179924.17361 \) (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

\(\Q(\sqrt{109}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{545}) \), 4.0.1295029.1, \(\Q(\sqrt{5}, \sqrt{109})\), 4.0.32375725.1, 4.4.13625.1 x2, 4.4.1485125.1 x2, 8.0.1048187569275625.1, 8.8.2205596265625.1, 8.0.26204689231890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$109$109.8.6.1$x^{8} - 5341 x^{4} + 15397776$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
109.8.6.1$x^{8} - 5341 x^{4} + 15397776$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$