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Properties

Label 16.0.60300391495...625.20
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{12}$
Root discriminant $96.89$
Ramified primes $5, 89$
Class number $11520$ (GRH)
Class group $[2, 24, 240]$ (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![41700391, -30739535, 29677943, -19110520, 18270660, -13507995, 13030788, -5203580, 1434649, -331650, 81672, -15095, 3620, -450, 97, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 97*x^14 - 450*x^13 + 3620*x^12 - 15095*x^11 + 81672*x^10 - 331650*x^9 + 1434649*x^8 - 5203580*x^7 + 13030788*x^6 - 13507995*x^5 + 18270660*x^4 - 19110520*x^3 + 29677943*x^2 - 30739535*x + 41700391)
 
gp: K = bnfinit(x^16 - 5*x^15 + 97*x^14 - 450*x^13 + 3620*x^12 - 15095*x^11 + 81672*x^10 - 331650*x^9 + 1434649*x^8 - 5203580*x^7 + 13030788*x^6 - 13507995*x^5 + 18270660*x^4 - 19110520*x^3 + 29677943*x^2 - 30739535*x + 41700391, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 97 x^{14} - 450 x^{13} + 3620 x^{12} - 15095 x^{11} + 81672 x^{10} - 331650 x^{9} + 1434649 x^{8} - 5203580 x^{7} + 13030788 x^{6} - 13507995 x^{5} + 18270660 x^{4} - 19110520 x^{3} + 29677943 x^{2} - 30739535 x + 41700391 \)

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:  \(60300391495425327222539306640625=5^{12}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.89$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{1068182} a^{14} - \frac{216359}{1068182} a^{13} + \frac{172851}{1068182} a^{12} + \frac{418945}{1068182} a^{11} + \frac{249205}{1068182} a^{10} + \frac{119113}{1068182} a^{9} + \frac{269665}{1068182} a^{8} + \frac{46719}{1068182} a^{7} + \frac{204615}{1068182} a^{6} - \frac{110919}{1068182} a^{5} + \frac{364467}{1068182} a^{4} - \frac{203297}{1068182} a^{3} - \frac{485403}{1068182} a^{2} - \frac{486563}{1068182} a - \frac{148245}{1068182}$, $\frac{1}{3248257640057523166185322608244882185476361403129353959582} a^{15} - \frac{1276263326434843379399982691531827957632062289535389}{3248257640057523166185322608244882185476361403129353959582} a^{14} + \frac{6338787172721014760183122514862276211386607234835698735}{29800528807867185010874519341696166839232673423205082198} a^{13} - \frac{91240276021589979006515238591733144165432864651217697641}{3248257640057523166185322608244882185476361403129353959582} a^{12} - \frac{1049001507820186032139246949440832759561513733719387350071}{3248257640057523166185322608244882185476361403129353959582} a^{11} - \frac{1557966969002904161331447293953639275721145782509212408295}{3248257640057523166185322608244882185476361403129353959582} a^{10} + \frac{528325403450261185795058272550772408189173861475559582365}{3248257640057523166185322608244882185476361403129353959582} a^{9} - \frac{680562968396712597391173155741171537961508940302207020099}{3248257640057523166185322608244882185476361403129353959582} a^{8} + \frac{677759802704635401188780592704182267184622447507980996631}{3248257640057523166185322608244882185476361403129353959582} a^{7} - \frac{354050593617414393837010444513775314183103067650346206663}{3248257640057523166185322608244882185476361403129353959582} a^{6} - \frac{372884908554603837535320618167620829382559337755472858291}{3248257640057523166185322608244882185476361403129353959582} a^{5} + \frac{549993351173302746108780188477240481396838174400835436111}{3248257640057523166185322608244882185476361403129353959582} a^{4} - \frac{95926859109092977941627860079641750025969649937157310817}{3248257640057523166185322608244882185476361403129353959582} a^{3} + \frac{2313350515627822440219674716272500570091006132545600861}{29800528807867185010874519341696166839232673423205082198} a^{2} + \frac{701367715236169385596488929150047033161638484187459939951}{3248257640057523166185322608244882185476361403129353959582} a + \frac{347732949108566742365157777948094900871568290846316929291}{1624128820028761583092661304122441092738180701564676979791}$

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

Class group and class number

$C_{2}\times C_{24}\times C_{240}$, which has order $11520$ (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:  \( 95443.616698 \) (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

\(\Q(\sqrt{445}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.39605.1 x2, 4.4.2225.1 x2, 4.0.88121125.1, 4.0.88121125.2, 8.8.39213900625.1, 8.0.7765332671265625.6, 8.0.7765332671265625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$