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Properties

Label 20.4.77062297047...7401.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 109^{6}$
Root discriminant $27.82$
Ramified primes $11, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T254

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 817, -2449, 5848, -11474, 12366, -4956, -7655, 21928, -29754, 29770, -23880, 15820, -8960, 4283, -1782, 625, -187, 45, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 45*x^18 - 187*x^17 + 625*x^16 - 1782*x^15 + 4283*x^14 - 8960*x^13 + 15820*x^12 - 23880*x^11 + 29770*x^10 - 29754*x^9 + 21928*x^8 - 7655*x^7 - 4956*x^6 + 12366*x^5 - 11474*x^4 + 5848*x^3 - 2449*x^2 + 817*x - 67)
 
gp: K = bnfinit(x^20 - 8*x^19 + 45*x^18 - 187*x^17 + 625*x^16 - 1782*x^15 + 4283*x^14 - 8960*x^13 + 15820*x^12 - 23880*x^11 + 29770*x^10 - 29754*x^9 + 21928*x^8 - 7655*x^7 - 4956*x^6 + 12366*x^5 - 11474*x^4 + 5848*x^3 - 2449*x^2 + 817*x - 67, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 45 x^{18} - 187 x^{17} + 625 x^{16} - 1782 x^{15} + 4283 x^{14} - 8960 x^{13} + 15820 x^{12} - 23880 x^{11} + 29770 x^{10} - 29754 x^{9} + 21928 x^{8} - 7655 x^{7} - 4956 x^{6} + 12366 x^{5} - 11474 x^{4} + 5848 x^{3} - 2449 x^{2} + 817 x - 67 \)

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:  \(77062297047310879021301897401=11^{16}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.82$
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:  $11, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{12118308993657455629913422887726314} a^{19} + \frac{840967629668184703377961699600110}{6059154496828727814956711443863157} a^{18} - \frac{716126419070592001752787272836509}{6059154496828727814956711443863157} a^{17} + \frac{2925772294820190456236994668319029}{12118308993657455629913422887726314} a^{16} - \frac{928023194651621691312434870594792}{6059154496828727814956711443863157} a^{15} - \frac{4868561272711264526980014979984495}{12118308993657455629913422887726314} a^{14} - \frac{4897365911625843009209947222397267}{12118308993657455629913422887726314} a^{13} + \frac{715399500461287702364163711379227}{12118308993657455629913422887726314} a^{12} + \frac{6041760874298151282762124452627555}{12118308993657455629913422887726314} a^{11} + \frac{2339346875973539329213284484240711}{6059154496828727814956711443863157} a^{10} + \frac{1168232862593707322235663400946973}{6059154496828727814956711443863157} a^{9} + \frac{4626105814060603457929622526468705}{12118308993657455629913422887726314} a^{8} - \frac{2360396485854997832033787992045819}{6059154496828727814956711443863157} a^{7} + \frac{5370690427086584661270672403189123}{12118308993657455629913422887726314} a^{6} - \frac{4748026619662654053380938776093521}{12118308993657455629913422887726314} a^{5} - \frac{2437874328471582084534101612995839}{6059154496828727814956711443863157} a^{4} - \frac{929505920445107708125502391008181}{6059154496828727814956711443863157} a^{3} - \frac{934661440332075472729979998162274}{6059154496828727814956711443863157} a^{2} + \frac{373421488629408296104584944573851}{12118308993657455629913422887726314} a + \frac{898668301903312472262695593604258}{6059154496828727814956711443863157}$

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:  \( 1951293.8897 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T254:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 28 conjugacy class representatives for t20n254
Character table for t20n254 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.23365118029.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
109Data not computed