↑

Properties

Label 20.8.24792226469...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{4}\cdot 5^{16}\cdot 6329^{5}$
Root discriminant $37.13$
Ramified primes $2, 5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-745, -1570, 9445, 780, -15795, -5000, 10670, 7135, -6084, -4496, 3662, 719, -639, -107, -164, 149, 23, -22, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 - 22*x^17 + 23*x^16 + 149*x^15 - 164*x^14 - 107*x^13 - 639*x^12 + 719*x^11 + 3662*x^10 - 4496*x^9 - 6084*x^8 + 7135*x^7 + 10670*x^6 - 5000*x^5 - 15795*x^4 + 780*x^3 + 9445*x^2 - 1570*x - 745)
 
gp: K = bnfinit(x^20 - 3*x^19 + 2*x^18 - 22*x^17 + 23*x^16 + 149*x^15 - 164*x^14 - 107*x^13 - 639*x^12 + 719*x^11 + 3662*x^10 - 4496*x^9 - 6084*x^8 + 7135*x^7 + 10670*x^6 - 5000*x^5 - 15795*x^4 + 780*x^3 + 9445*x^2 - 1570*x - 745, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 2 x^{18} - 22 x^{17} + 23 x^{16} + 149 x^{15} - 164 x^{14} - 107 x^{13} - 639 x^{12} + 719 x^{11} + 3662 x^{10} - 4496 x^{9} - 6084 x^{8} + 7135 x^{7} + 10670 x^{6} - 5000 x^{5} - 15795 x^{4} + 780 x^{3} + 9445 x^{2} - 1570 x - 745 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24792226469882721311035156250000=2^{4}\cdot 5^{16}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.13$
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:  $2, 5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{274669024975229959928596636988092095549499} a^{19} - \frac{83993267524199933220122407569754088505880}{274669024975229959928596636988092095549499} a^{18} - \frac{117055067862647988103419378411761274958912}{274669024975229959928596636988092095549499} a^{17} - \frac{44753796333038357861360629729318931192570}{274669024975229959928596636988092095549499} a^{16} + \frac{113004352712947641483167611030052409526869}{274669024975229959928596636988092095549499} a^{15} - \frac{12945473678848831876178939168183549398788}{274669024975229959928596636988092095549499} a^{14} - \frac{61317936725750731703258554686961332420812}{274669024975229959928596636988092095549499} a^{13} - \frac{86988653398254004391890763287560220962846}{274669024975229959928596636988092095549499} a^{12} + \frac{2007549302507384162300014862816266300010}{274669024975229959928596636988092095549499} a^{11} - \frac{124854332260215562653459242556438097866984}{274669024975229959928596636988092095549499} a^{10} - \frac{66545921984393675907938163054908408360162}{274669024975229959928596636988092095549499} a^{9} + \frac{41632749461920503685046367077591463680014}{274669024975229959928596636988092095549499} a^{8} + \frac{110249953741913638136690541209792387596729}{274669024975229959928596636988092095549499} a^{7} - \frac{122122602768622591273451721388754714991118}{274669024975229959928596636988092095549499} a^{6} - \frac{40328737498531439791590273203764933796615}{274669024975229959928596636988092095549499} a^{5} + \frac{115483197431544618835256268024099360450345}{274669024975229959928596636988092095549499} a^{4} + \frac{24403637645117087430678403708770343436389}{274669024975229959928596636988092095549499} a^{3} + \frac{103877208126513546029987814827532863602060}{274669024975229959928596636988092095549499} a^{2} - \frac{12344595041630025864425846177275017280990}{274669024975229959928596636988092095549499} a + \frac{88837308957393513620730057895989639702391}{274669024975229959928596636988092095549499}$

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:  $13$
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:  \( 155355946.232 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
6329Data not computed