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Properties

Label 20.0.33359761956...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 7^{10}$
Root discriminant $26.68$
Ramified primes $2, 3, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2025, 2025, 2025, 6075, -540, 765, 6885, -1065, 2326, -2473, 3759, -1983, 1062, -707, 509, -379, 204, -81, 27, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 27*x^18 - 81*x^17 + 204*x^16 - 379*x^15 + 509*x^14 - 707*x^13 + 1062*x^12 - 1983*x^11 + 3759*x^10 - 2473*x^9 + 2326*x^8 - 1065*x^7 + 6885*x^6 + 765*x^5 - 540*x^4 + 6075*x^3 + 2025*x^2 + 2025*x + 2025)
 
gp: K = bnfinit(x^20 - 5*x^19 + 27*x^18 - 81*x^17 + 204*x^16 - 379*x^15 + 509*x^14 - 707*x^13 + 1062*x^12 - 1983*x^11 + 3759*x^10 - 2473*x^9 + 2326*x^8 - 1065*x^7 + 6885*x^6 + 765*x^5 - 540*x^4 + 6075*x^3 + 2025*x^2 + 2025*x + 2025, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 27 x^{18} - 81 x^{17} + 204 x^{16} - 379 x^{15} + 509 x^{14} - 707 x^{13} + 1062 x^{12} - 1983 x^{11} + 3759 x^{10} - 2473 x^{9} + 2326 x^{8} - 1065 x^{7} + 6885 x^{6} + 765 x^{5} - 540 x^{4} + 6075 x^{3} + 2025 x^{2} + 2025 x + 2025 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33359761956402000000000000000=2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.68$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{12} - \frac{2}{15} a^{11} + \frac{2}{15} a^{10} + \frac{1}{3} a^{9} + \frac{1}{15} a^{8} - \frac{2}{15} a^{6} + \frac{4}{15} a^{5} - \frac{4}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{7}{15} a^{8} + \frac{1}{5} a^{7} - \frac{1}{15} a^{5} + \frac{2}{15} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{14} - \frac{1}{5} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{7}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{15} + \frac{2}{15} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{2}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{45} a^{16} + \frac{1}{45} a^{15} + \frac{2}{45} a^{11} + \frac{2}{45} a^{10} + \frac{2}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{15} a^{6} + \frac{2}{45} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{2475} a^{17} - \frac{7}{825} a^{16} - \frac{76}{2475} a^{15} - \frac{7}{825} a^{14} - \frac{2}{75} a^{13} + \frac{71}{2475} a^{12} + \frac{14}{825} a^{11} + \frac{287}{2475} a^{10} - \frac{53}{225} a^{9} + \frac{74}{825} a^{8} - \frac{119}{275} a^{7} - \frac{224}{495} a^{6} + \frac{4}{495} a^{5} + \frac{1}{9} a^{4} + \frac{26}{55} a^{3} + \frac{5}{33} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{51975} a^{18} + \frac{4}{51975} a^{17} + \frac{38}{17325} a^{16} - \frac{457}{17325} a^{15} + \frac{74}{2475} a^{14} + \frac{71}{51975} a^{13} + \frac{236}{7425} a^{12} + \frac{3922}{51975} a^{11} + \frac{1354}{17325} a^{10} + \frac{4804}{17325} a^{9} - \frac{487}{17325} a^{8} - \frac{632}{1485} a^{7} + \frac{2951}{10395} a^{6} + \frac{9}{385} a^{5} + \frac{1123}{3465} a^{4} + \frac{17}{77} a^{3} + \frac{86}{231} a^{2} - \frac{12}{77}$, $\frac{1}{2144842159642164927909951525} a^{19} - \frac{372929307455703044947}{142989477309477661860663435} a^{18} - \frac{73993458463963721845817}{428968431928432985581990305} a^{17} + \frac{1930518022412008773953782}{714947386547388309303317175} a^{16} - \frac{9985577183137147861794143}{714947386547388309303317175} a^{15} - \frac{19269842997898080711441457}{2144842159642164927909951525} a^{14} - \frac{6072807572564436090858628}{714947386547388309303317175} a^{13} + \frac{45761747515730161784633711}{2144842159642164927909951525} a^{12} - \frac{144049373517116160103796242}{2144842159642164927909951525} a^{11} - \frac{94565103133131100090152319}{714947386547388309303317175} a^{10} + \frac{3791809946856954829413584}{28597895461895532372132687} a^{9} - \frac{486824751689692018972300537}{2144842159642164927909951525} a^{8} + \frac{319628689444875800697711968}{2144842159642164927909951525} a^{7} + \frac{83414422235997058839333607}{428968431928432985581990305} a^{6} + \frac{58316327361081759946969967}{142989477309477661860663435} a^{5} + \frac{15996126017826483069462169}{142989477309477661860663435} a^{4} + \frac{253094617578335253954307}{47663159103159220620221145} a^{3} + \frac{2936492526921336917480740}{9532631820631844124044229} a^{2} - \frac{903835579719580005531248}{3177543940210614708014743} a + \frac{1147325030941349191878621}{3177543940210614708014743}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 737205.625442 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.55125.1, 5.1.882000.1 x5, 10.2.3889620000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.882000.1
Degree 10 sibling: 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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$