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Properties

Label 20.0.46398098569...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 3169^{2}\cdot 8951^{2}$
Root discriminant $21.54$
Ramified primes $3, 5, 3169, 8951$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 67, 26, 205, 31, 475, 9, 467, 25, 348, 41, 122, 54, 35, 24, 2, 8, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 8*x^17 + 2*x^16 + 24*x^15 + 35*x^14 + 54*x^13 + 122*x^12 + 41*x^11 + 348*x^10 + 25*x^9 + 467*x^8 + 9*x^7 + 475*x^6 + 31*x^5 + 205*x^4 + 26*x^3 + 67*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 + 8*x^17 + 2*x^16 + 24*x^15 + 35*x^14 + 54*x^13 + 122*x^12 + 41*x^11 + 348*x^10 + 25*x^9 + 467*x^8 + 9*x^7 + 475*x^6 + 31*x^5 + 205*x^4 + 26*x^3 + 67*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} + 8 x^{17} + 2 x^{16} + 24 x^{15} + 35 x^{14} + 54 x^{13} + 122 x^{12} + 41 x^{11} + 348 x^{10} + 25 x^{9} + 467 x^{8} + 9 x^{7} + 475 x^{6} + 31 x^{5} + 205 x^{4} + 26 x^{3} + 67 x^{2} + 8 x + 1 \)

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:  \(463980985698590430556640625=3^{10}\cdot 5^{10}\cdot 3169^{2}\cdot 8951^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.54$
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:  $3, 5, 3169, 8951$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{107} a^{18} - \frac{9}{107} a^{17} + \frac{11}{107} a^{16} - \frac{39}{107} a^{15} + \frac{38}{107} a^{14} - \frac{31}{107} a^{13} - \frac{9}{107} a^{12} - \frac{30}{107} a^{11} - \frac{25}{107} a^{10} + \frac{21}{107} a^{9} - \frac{39}{107} a^{8} - \frac{44}{107} a^{7} - \frac{2}{107} a^{6} - \frac{48}{107} a^{5} + \frac{24}{107} a^{4} + \frac{22}{107} a^{3} - \frac{9}{107} a^{2} - \frac{26}{107} a - \frac{5}{107}$, $\frac{1}{1013269627646835959} a^{19} + \frac{2203196195743990}{1013269627646835959} a^{18} + \frac{299207928771039095}{1013269627646835959} a^{17} - \frac{173155842238798840}{1013269627646835959} a^{16} - \frac{369226622267909557}{1013269627646835959} a^{15} + \frac{328049987972785143}{1013269627646835959} a^{14} + \frac{408972720611469001}{1013269627646835959} a^{13} - \frac{316132906669086950}{1013269627646835959} a^{12} - \frac{9246360604861706}{1013269627646835959} a^{11} - \frac{311382318186184234}{1013269627646835959} a^{10} - \frac{476731929159332102}{1013269627646835959} a^{9} - \frac{14716137871415567}{1013269627646835959} a^{8} + \frac{128273252490164244}{1013269627646835959} a^{7} - \frac{219007867424932074}{1013269627646835959} a^{6} - \frac{232626409694181400}{1013269627646835959} a^{5} - \frac{3348976271290444}{9469809604176037} a^{4} + \frac{441839778832778361}{1013269627646835959} a^{3} + \frac{252447238450825333}{1013269627646835959} a^{2} - \frac{502232594466684399}{1013269627646835959} a - \frac{294803707342316480}{1013269627646835959}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{1187788991647979}{9469809604176037} a^{19} + \frac{1135660668753794}{9469809604176037} a^{18} - \frac{3488469199552919}{9469809604176037} a^{17} - \frac{9612454231269580}{9469809604176037} a^{16} - \frac{2801845486365625}{9469809604176037} a^{15} - \frac{28241855703647755}{9469809604176037} a^{14} - \frac{42225764662064929}{9469809604176037} a^{13} - \frac{65418895483406427}{9469809604176037} a^{12} - \frac{145556889136975242}{9469809604176037} a^{11} - \frac{51636313862465392}{9469809604176037} a^{10} - \frac{409809539762363151}{9469809604176037} a^{9} - \frac{40054750177890931}{9469809604176037} a^{8} - \frac{547387738586789695}{9469809604176037} a^{7} - \frac{13362445947532206}{9469809604176037} a^{6} - \frac{555212014385831572}{9469809604176037} a^{5} - \frac{37053007185599963}{9469809604176037} a^{4} - \frac{237193833731899091}{9469809604176037} a^{3} - \frac{11096763973838309}{9469809604176037} a^{2} - \frac{77025798468989738}{9469809604176037} a + \frac{275343774430485}{9469809604176037} \) (order $6$)
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:  \( 71627.8364606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.8.88642871875.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
3Data not computed
5Data not computed
3169Data not computed
8951Data not computed