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Properties

Label 20.8.76237898330...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $31.20$
Ramified primes $5, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 50, -91, -147, 7, 252, 135, 994, -1293, -929, 1655, -1673, 1847, -1544, 1190, -629, 207, -68, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 17*x^18 - 68*x^17 + 207*x^16 - 629*x^15 + 1190*x^14 - 1544*x^13 + 1847*x^12 - 1673*x^11 + 1655*x^10 - 929*x^9 - 1293*x^8 + 994*x^7 + 135*x^6 + 252*x^5 + 7*x^4 - 147*x^3 - 91*x^2 + 50*x - 5)
 
gp: K = bnfinit(x^20 - 5*x^19 + 17*x^18 - 68*x^17 + 207*x^16 - 629*x^15 + 1190*x^14 - 1544*x^13 + 1847*x^12 - 1673*x^11 + 1655*x^10 - 929*x^9 - 1293*x^8 + 994*x^7 + 135*x^6 + 252*x^5 + 7*x^4 - 147*x^3 - 91*x^2 + 50*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 17 x^{18} - 68 x^{17} + 207 x^{16} - 629 x^{15} + 1190 x^{14} - 1544 x^{13} + 1847 x^{12} - 1673 x^{11} + 1655 x^{10} - 929 x^{9} - 1293 x^{8} + 994 x^{7} + 135 x^{6} + 252 x^{5} + 7 x^{4} - 147 x^{3} - 91 x^{2} + 50 x - 5 \)

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:  \(762378983303206514165048828125=5^{10}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.20$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2427088373034832308030795162586681} a^{19} - \frac{320146873847078525857877297617652}{2427088373034832308030795162586681} a^{18} + \frac{569994951216933729235118100289769}{2427088373034832308030795162586681} a^{17} + \frac{431021873978031771994291320977929}{2427088373034832308030795162586681} a^{16} - \frac{505369080792275039188905486052427}{2427088373034832308030795162586681} a^{15} + \frac{416904477094762500915181412682082}{2427088373034832308030795162586681} a^{14} + \frac{881724344147416867185410189099978}{2427088373034832308030795162586681} a^{13} + \frac{382306527497623456676975058757317}{2427088373034832308030795162586681} a^{12} + \frac{1193418866671580479979809966978723}{2427088373034832308030795162586681} a^{11} + \frac{846769547212696647177078385178129}{2427088373034832308030795162586681} a^{10} + \frac{871646006367710745275673962429528}{2427088373034832308030795162586681} a^{9} + \frac{31947355067504253006362528023698}{2427088373034832308030795162586681} a^{8} - \frac{1011794448179901025200712868866779}{2427088373034832308030795162586681} a^{7} + \frac{479773634042929163098645354366967}{2427088373034832308030795162586681} a^{6} + \frac{263344450033974896482723347382353}{2427088373034832308030795162586681} a^{5} + \frac{986892477614228241012658286829778}{2427088373034832308030795162586681} a^{4} + \frac{947064463233645189273510172909166}{2427088373034832308030795162586681} a^{3} - \frac{411407968129235793725223539443414}{2427088373034832308030795162586681} a^{2} - \frac{996743915707675522231488829839606}{2427088373034832308030795162586681} a - \frac{385005193403642145766148393303082}{2427088373034832308030795162586681}$

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:  \( 28113569.3771 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61Data not computed
397Data not computed