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Properties

Label 20.4.18835106651...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{11}\cdot 19^{4}\cdot 461^{4}$
Root discriminant $25.93$
Ramified primes $2, 5, 19, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 10, -4, -143, -183, 70, 1084, 8, 222, 532, -536, 428, -197, -66, 146, -151, 100, -50, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 - 50*x^17 + 100*x^16 - 151*x^15 + 146*x^14 - 66*x^13 - 197*x^12 + 428*x^11 - 536*x^10 + 532*x^9 + 222*x^8 + 8*x^7 + 1084*x^6 + 70*x^5 - 183*x^4 - 143*x^3 - 4*x^2 + 10*x + 5)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 - 50*x^17 + 100*x^16 - 151*x^15 + 146*x^14 - 66*x^13 - 197*x^12 + 428*x^11 - 536*x^10 + 532*x^9 + 222*x^8 + 8*x^7 + 1084*x^6 + 70*x^5 - 183*x^4 - 143*x^3 - 4*x^2 + 10*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} - 50 x^{17} + 100 x^{16} - 151 x^{15} + 146 x^{14} - 66 x^{13} - 197 x^{12} + 428 x^{11} - 536 x^{10} + 532 x^{9} + 222 x^{8} + 8 x^{7} + 1084 x^{6} + 70 x^{5} - 183 x^{4} - 143 x^{3} - 4 x^{2} + 10 x + 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18835106651668995200000000000=2^{16}\cdot 5^{11}\cdot 19^{4}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.93$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{15} a^{18} - \frac{2}{15} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{7}{15} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{4}{15} a^{9} + \frac{1}{3} a^{8} + \frac{1}{5} a^{7} + \frac{1}{3} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{7}{15} a^{3} + \frac{2}{15} a^{2} - \frac{1}{3}$, $\frac{1}{19134431784264120857205} a^{19} + \frac{103544232335757477469}{3826886356852824171441} a^{18} + \frac{385762685935697691077}{6378143928088040285735} a^{17} + \frac{3428312087527647630859}{19134431784264120857205} a^{16} - \frac{5819588689082430763306}{19134431784264120857205} a^{15} - \frac{2099930220093510742228}{6378143928088040285735} a^{14} + \frac{17036726377865115022}{3826886356852824171441} a^{13} + \frac{5787491301724318388069}{19134431784264120857205} a^{12} - \frac{5482628143231116697253}{19134431784264120857205} a^{11} - \frac{8028681198867729715759}{19134431784264120857205} a^{10} - \frac{4814323521954672478358}{19134431784264120857205} a^{9} + \frac{415606703902351011154}{19134431784264120857205} a^{8} + \frac{4816021825453026475316}{19134431784264120857205} a^{7} - \frac{1415036596752305201266}{3826886356852824171441} a^{6} - \frac{600345560959730798092}{1275628785617608057147} a^{5} - \frac{6826820959204057663508}{19134431784264120857205} a^{4} + \frac{2367653673116192681473}{19134431784264120857205} a^{3} - \frac{152815823715971061497}{19134431784264120857205} a^{2} - \frac{773618765172456239638}{3826886356852824171441} a - \frac{1850326441501645071275}{3826886356852824171441}$

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:  $11$
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:  \( 3187787.71177 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed