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Properties

Label 20.0.22655426006...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 211^{8}$
Root discriminant $32.94$
Ramified primes $3, 5, 211$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $C_{10}\times D_5$ (as 20T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 20, -47, 245, -526, 1420, -1573, 2627, -1998, 3161, -1794, 2040, -790, 844, -274, 171, -28, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^18 - 28*x^17 + 171*x^16 - 274*x^15 + 844*x^14 - 790*x^13 + 2040*x^12 - 1794*x^11 + 3161*x^10 - 1998*x^9 + 2627*x^8 - 1573*x^7 + 1420*x^6 - 526*x^5 + 245*x^4 - 47*x^3 + 20*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^18 - 28*x^17 + 171*x^16 - 274*x^15 + 844*x^14 - 790*x^13 + 2040*x^12 - 1794*x^11 + 3161*x^10 - 1998*x^9 + 2627*x^8 - 1573*x^7 + 1420*x^6 - 526*x^5 + 245*x^4 - 47*x^3 + 20*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{18} - 28 x^{17} + 171 x^{16} - 274 x^{15} + 844 x^{14} - 790 x^{13} + 2040 x^{12} - 1794 x^{11} + 3161 x^{10} - 1998 x^{9} + 2627 x^{8} - 1573 x^{7} + 1420 x^{6} - 526 x^{5} + 245 x^{4} - 47 x^{3} + 20 x^{2} - 3 x + 1 \)

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:  \(2265542600600196424321962890625=3^{10}\cdot 5^{10}\cdot 211^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.94$
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:  $3, 5, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{179} a^{18} - \frac{8}{179} a^{17} + \frac{28}{179} a^{16} - \frac{87}{179} a^{15} + \frac{43}{179} a^{14} - \frac{85}{179} a^{13} - \frac{15}{179} a^{12} + \frac{33}{179} a^{11} + \frac{55}{179} a^{10} + \frac{89}{179} a^{9} - \frac{69}{179} a^{8} + \frac{45}{179} a^{7} + \frac{8}{179} a^{6} - \frac{19}{179} a^{5} - \frac{7}{179} a^{4} + \frac{21}{179} a^{3} + \frac{13}{179} a^{2} + \frac{14}{179} a + \frac{5}{179}$, $\frac{1}{2177493124667658664610827} a^{19} - \frac{2363504313826731809060}{2177493124667658664610827} a^{18} - \frac{190874666038040339089228}{2177493124667658664610827} a^{17} + \frac{199012385228443520117328}{2177493124667658664610827} a^{16} - \frac{600952372753077944273261}{2177493124667658664610827} a^{15} - \frac{917023905035553181874}{27563204109717198286213} a^{14} - \frac{132085780659592686549313}{2177493124667658664610827} a^{13} - \frac{910641016133525281975603}{2177493124667658664610827} a^{12} + \frac{327826331463166730039843}{2177493124667658664610827} a^{11} + \frac{559156467864685234252584}{2177493124667658664610827} a^{10} - \frac{547418926986445178887971}{2177493124667658664610827} a^{9} + \frac{517928168743405042077695}{2177493124667658664610827} a^{8} - \frac{902158399847573227290012}{2177493124667658664610827} a^{7} - \frac{450019235267396946367868}{2177493124667658664610827} a^{6} + \frac{686793799807171534655236}{2177493124667658664610827} a^{5} - \frac{504676735536172461751389}{2177493124667658664610827} a^{4} - \frac{757509139332009747285330}{2177493124667658664610827} a^{3} + \frac{773560521721769919786709}{2177493124667658664610827} a^{2} + \frac{39873007161090878425075}{2177493124667658664610827} a + \frac{884467571312362311632873}{2177493124667658664610827}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( \frac{2898309589442693763770}{12164766059595858461513} a^{19} - \frac{5315238256937966297504}{12164766059595858461513} a^{18} + \frac{45244681359600409162699}{12164766059595858461513} a^{17} - \frac{73072451980768621659006}{12164766059595858461513} a^{16} + \frac{479417576734164484616042}{12164766059595858461513} a^{15} - \frac{8943055300991463881343}{153984380501213398247} a^{14} + \frac{2285124978576323599828412}{12164766059595858461513} a^{13} - \frac{1830954274831398638382440}{12164766059595858461513} a^{12} + \frac{5384770870595414502565379}{12164766059595858461513} a^{11} - \frac{4056762247911081075850598}{12164766059595858461513} a^{10} + \frac{7950061612415443828592718}{12164766059595858461513} a^{9} - \frac{3907801434610805837198923}{12164766059595858461513} a^{8} + \frac{6109410827582731864095628}{12164766059595858461513} a^{7} - \frac{2883553998208237372351966}{12164766059595858461513} a^{6} + \frac{2935085734585437926666882}{12164766059595858461513} a^{5} - \frac{537251106504941092198094}{12164766059595858461513} a^{4} + \frac{232705235773807887707516}{12164766059595858461513} a^{3} + \frac{87793722487717720088834}{12164766059595858461513} a^{2} + \frac{16951095734147916946298}{12164766059595858461513} a + \frac{9995551780264801958597}{12164766059595858461513} \) (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:  \( 454775.127492 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}\times D_5$ (as 20T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.10.6194123253125.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
211Data not computed