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Properties

Label 20.0.19108974585...0761.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}$
Root discriminant $20.61$
Ramified primes $3, 61, 97, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 25, 30, 72, -15, 124, -111, 216, -295, 273, -319, 310, -257, 186, -123, 72, -38, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 17*x^18 - 38*x^17 + 72*x^16 - 123*x^15 + 186*x^14 - 257*x^13 + 310*x^12 - 319*x^11 + 273*x^10 - 295*x^9 + 216*x^8 - 111*x^7 + 124*x^6 - 15*x^5 + 72*x^4 + 30*x^3 + 25*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 17*x^18 - 38*x^17 + 72*x^16 - 123*x^15 + 186*x^14 - 257*x^13 + 310*x^12 - 319*x^11 + 273*x^10 - 295*x^9 + 216*x^8 - 111*x^7 + 124*x^6 - 15*x^5 + 72*x^4 + 30*x^3 + 25*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 17 x^{18} - 38 x^{17} + 72 x^{16} - 123 x^{15} + 186 x^{14} - 257 x^{13} + 310 x^{12} - 319 x^{11} + 273 x^{10} - 295 x^{9} + 216 x^{8} - 111 x^{7} + 124 x^{6} - 15 x^{5} + 72 x^{4} + 30 x^{3} + 25 x^{2} + 5 x + 1 \)

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:  \(191089745854193889068530761=3^{10}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.61$
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:  $3, 61, 97, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} - \frac{1}{13} a^{17} - \frac{3}{13} a^{16} + \frac{5}{13} a^{15} - \frac{3}{13} a^{14} + \frac{6}{13} a^{13} - \frac{2}{13} a^{12} + \frac{3}{13} a^{11} + \frac{3}{13} a^{10} - \frac{4}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{6}{13} a^{6} - \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} + \frac{5}{13} a - \frac{4}{13}$, $\frac{1}{29919870926736635} a^{19} + \frac{832056392929964}{29919870926736635} a^{18} + \frac{5510023652282918}{29919870926736635} a^{17} - \frac{2268763510051096}{29919870926736635} a^{16} - \frac{5976978775457692}{29919870926736635} a^{15} - \frac{292132114161126}{29919870926736635} a^{14} + \frac{4736658302451082}{29919870926736635} a^{13} + \frac{728555546440442}{2301528532825895} a^{12} + \frac{9865798788375804}{29919870926736635} a^{11} - \frac{3927254424388158}{29919870926736635} a^{10} + \frac{1494128716133061}{29919870926736635} a^{9} + \frac{2230803041979384}{29919870926736635} a^{8} + \frac{14137833083750647}{29919870926736635} a^{7} - \frac{8739441228745418}{29919870926736635} a^{6} + \frac{9144196150353237}{29919870926736635} a^{5} - \frac{6357509607003777}{29919870926736635} a^{4} - \frac{4906083002075421}{29919870926736635} a^{3} + \frac{13291050284898526}{29919870926736635} a^{2} + \frac{3360870223197289}{29919870926736635} a + \frac{10784806387239151}{29919870926736635}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{395026395648133}{2301528532825895} a^{19} - \frac{1966844656854868}{2301528532825895} a^{18} + \frac{6702067012055614}{2301528532825895} a^{17} - \frac{15025823028784033}{2301528532825895} a^{16} + \frac{28672863166768614}{2301528532825895} a^{15} - \frac{49270852787167068}{2301528532825895} a^{14} + \frac{74910926211655671}{2301528532825895} a^{13} - \frac{104207299815215342}{2301528532825895} a^{12} + \frac{126911014793231792}{2301528532825895} a^{11} - \frac{132580934485479324}{2301528532825895} a^{10} + \frac{116501876273284348}{2301528532825895} a^{9} - \frac{126132795684327368}{2301528532825895} a^{8} + \frac{93477622651990126}{2301528532825895} a^{7} - \frac{52782638309758904}{2301528532825895} a^{6} + \frac{57719664718384846}{2301528532825895} a^{5} - \frac{10036236471743576}{2301528532825895} a^{4} + \frac{32025747783963752}{2301528532825895} a^{3} + \frac{7958853620659238}{2301528532825895} a^{2} + \frac{11774037744196157}{2301528532825895} a + \frac{2370930269132673}{2301528532825895} \) (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:  \( 137168.028568 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.24217.1, 10.6.56886919633.1, 10.4.13823521470819.1, 10.0.142510530627.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
97Data not computed
397Data not computed