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Properties

Label 20.4.41153942158...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $42.73$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3044, -21984, -76620, -164952, -99956, 43972, 172388, -60716, 50540, 18552, -8586, 11252, -1298, 1290, 296, 48, 106, -14, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 18*x^18 - 14*x^17 + 106*x^16 + 48*x^15 + 296*x^14 + 1290*x^13 - 1298*x^12 + 11252*x^11 - 8586*x^10 + 18552*x^9 + 50540*x^8 - 60716*x^7 + 172388*x^6 + 43972*x^5 - 99956*x^4 - 164952*x^3 - 76620*x^2 - 21984*x - 3044)
 
gp: K = bnfinit(x^20 - 2*x^19 + 18*x^18 - 14*x^17 + 106*x^16 + 48*x^15 + 296*x^14 + 1290*x^13 - 1298*x^12 + 11252*x^11 - 8586*x^10 + 18552*x^9 + 50540*x^8 - 60716*x^7 + 172388*x^6 + 43972*x^5 - 99956*x^4 - 164952*x^3 - 76620*x^2 - 21984*x - 3044, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 18 x^{18} - 14 x^{17} + 106 x^{16} + 48 x^{15} + 296 x^{14} + 1290 x^{13} - 1298 x^{12} + 11252 x^{11} - 8586 x^{10} + 18552 x^{9} + 50540 x^{8} - 60716 x^{7} + 172388 x^{6} + 43972 x^{5} - 99956 x^{4} - 164952 x^{3} - 76620 x^{2} - 21984 x - 3044 \)

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:  \(411539421581712055500800000000000=2^{24}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.73$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{27296592439892580596147823079108712471896798150706} a^{19} - \frac{4027291467901048501099747608931230611129386170833}{27296592439892580596147823079108712471896798150706} a^{18} - \frac{2855812220574125172499367204210968794848873333088}{13648296219946290298073911539554356235948399075353} a^{17} + \frac{3846116597217494505036643612824898807159056348741}{27296592439892580596147823079108712471896798150706} a^{16} + \frac{1199065365723553515693727755803037324387458571705}{13648296219946290298073911539554356235948399075353} a^{15} + \frac{615413241944608590958274669303299611735614711319}{27296592439892580596147823079108712471896798150706} a^{14} + \frac{5967991020171034182956006735837487866894031517783}{27296592439892580596147823079108712471896798150706} a^{13} - \frac{723595597843008751368208297280752663908459310889}{13648296219946290298073911539554356235948399075353} a^{12} + \frac{5225895244430915780192501891953844802515163992467}{27296592439892580596147823079108712471896798150706} a^{11} + \frac{2820779365978315452161943242618256062933026771671}{27296592439892580596147823079108712471896798150706} a^{10} + \frac{3053049588815887452017693491835979010165545706675}{13648296219946290298073911539554356235948399075353} a^{9} + \frac{3252869101802794342975734716602124382101796269627}{13648296219946290298073911539554356235948399075353} a^{8} - \frac{6212884297229027623154232156801347612625961694244}{13648296219946290298073911539554356235948399075353} a^{7} - \frac{1680765537453487197415161193529089053191419279061}{13648296219946290298073911539554356235948399075353} a^{6} + \frac{3972250330411821294423554760451521552101041708175}{13648296219946290298073911539554356235948399075353} a^{5} + \frac{1065079314897757358559454001149343131103563669991}{13648296219946290298073911539554356235948399075353} a^{4} + \frac{956647691592310590746261224775000253645162919217}{13648296219946290298073911539554356235948399075353} a^{3} + \frac{5597748372409715940344426393499502993488385769308}{13648296219946290298073911539554356235948399075353} a^{2} + \frac{3533598042426838819305187746623783820961000992220}{13648296219946290298073911539554356235948399075353} a + \frac{4982505714569551203662676530568329433546373946969}{13648296219946290298073911539554356235948399075353}$

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:  \( 252236239.947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed