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Properties

Label 20.16.3956448137...6016.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{8}$
Root discriminant $95.47$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![978121, 0, -595654, 0, -2229321, 0, 2802780, 0, -874464, 0, -175704, 0, 151221, 0, -29870, 0, 2341, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 80*x^18 + 2341*x^16 - 29870*x^14 + 151221*x^12 - 175704*x^10 - 874464*x^8 + 2802780*x^6 - 2229321*x^4 - 595654*x^2 + 978121)
 
gp: K = bnfinit(x^20 - 80*x^18 + 2341*x^16 - 29870*x^14 + 151221*x^12 - 175704*x^10 - 874464*x^8 + 2802780*x^6 - 2229321*x^4 - 595654*x^2 + 978121, 1)
 

Normalized defining polynomial

\( x^{20} - 80 x^{18} + 2341 x^{16} - 29870 x^{14} + 151221 x^{12} - 175704 x^{10} - 874464 x^{8} + 2802780 x^{6} - 2229321 x^{4} - 595654 x^{2} + 978121 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3956448137628099441486726576701509206016=2^{40}\cdot 11^{16}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.47$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{4}{11} a^{8} + \frac{2}{11} a^{6} - \frac{5}{11} a^{4} - \frac{2}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{4}{11} a^{9} + \frac{2}{11} a^{7} - \frac{5}{11} a^{5} - \frac{2}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{3}{11} a^{8} - \frac{2}{11} a^{6} - \frac{4}{11} a^{4} - \frac{2}{11} a^{2} - \frac{4}{11}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{9} - \frac{2}{11} a^{7} - \frac{4}{11} a^{5} - \frac{2}{11} a^{3} - \frac{4}{11} a$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{8} + \frac{2}{11} a^{6} + \frac{5}{11} a^{4} + \frac{1}{11} a^{2} + \frac{3}{11}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{9} + \frac{2}{11} a^{7} + \frac{5}{11} a^{5} + \frac{1}{11} a^{3} + \frac{3}{11} a$, $\frac{1}{2783} a^{16} + \frac{35}{2783} a^{14} + \frac{87}{2783} a^{12} - \frac{39}{2783} a^{10} + \frac{962}{2783} a^{8} - \frac{582}{2783} a^{6} + \frac{502}{2783} a^{4} - \frac{38}{121} a^{2} + \frac{16}{121}$, $\frac{1}{2783} a^{17} + \frac{35}{2783} a^{15} + \frac{87}{2783} a^{13} - \frac{39}{2783} a^{11} + \frac{962}{2783} a^{9} - \frac{582}{2783} a^{7} + \frac{502}{2783} a^{5} - \frac{38}{121} a^{3} + \frac{16}{121} a$, $\frac{1}{1275533101463039613073} a^{18} - \frac{172751591928006559}{1275533101463039613073} a^{16} - \frac{116603827456641366}{55457960933175635351} a^{14} + \frac{44289658927978667689}{1275533101463039613073} a^{12} - \frac{34180345914344235685}{1275533101463039613073} a^{10} - \frac{159516080666140524211}{1275533101463039613073} a^{8} + \frac{412506539739736243163}{1275533101463039613073} a^{6} - \frac{392370232145118911327}{1275533101463039613073} a^{4} + \frac{7770210750524868801}{55457960933175635351} a^{2} + \frac{11528406528821476584}{55457960933175635351}$, $\frac{1}{54847923362910703362139} a^{19} - \frac{62033352745994055}{453288622833972755059} a^{17} - \frac{1650837487334805104494}{54847923362910703362139} a^{15} - \frac{2449026931786694524951}{54847923362910703362139} a^{13} + \frac{599690397644183893988}{54847923362910703362139} a^{11} - \frac{26926919670908483133873}{54847923362910703362139} a^{9} - \frac{15380179079057229602204}{54847923362910703362139} a^{7} + \frac{20161450082303322140868}{54847923362910703362139} a^{5} - \frac{419393587511621347291}{2384692320126552320093} a^{3} - \frac{1114130701337950097152}{2384692320126552320093} a$

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:  $17$
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:  \( 13041299003600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$