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Properties

Label 20.0.357...081.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.575\times 10^{21}$
Root discriminant $11.96$
Ramified primes $3, 7, 11$
Class number $1$
Class group trivial
Galois group $C_{10}\times D_5$ (as 20T24)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 2*x^18 + 3*x^17 - 2*x^16 - 9*x^15 + 18*x^14 - 13*x^13 - x^12 + 6*x^11 + 10*x^10 - 26*x^9 + 20*x^8 - 6*x^7 + x^6 + 5*x^5 - 10*x^4 + 6*x^3 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 6, -10, 5, 1, -6, 20, -26, 10, 6, -1, -13, 18, -9, -2, 3, 2, -3, 1]);
 

\( x^{20} - 3 x^{19} + 2 x^{18} + 3 x^{17} - 2 x^{16} - 9 x^{15} + 18 x^{14} - 13 x^{13} - x^{12} + 6 x^{11} + 10 x^{10} - 26 x^{9} + 20 x^{8} - 6 x^{7} + x^{6} + 5 x^{5} - 10 x^{4} + 6 x^{3} - 2 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3575480621700351753081\)\(\medspace = 3^{10}\cdot 7^{10}\cdot 11^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.96$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2663593} a^{19} - \frac{605577}{2663593} a^{18} + \frac{865553}{2663593} a^{17} - \frac{1243914}{2663593} a^{16} - \frac{768917}{2663593} a^{15} + \frac{133054}{2663593} a^{14} - \frac{354728}{2663593} a^{13} + \frac{605595}{2663593} a^{12} - \frac{1111512}{2663593} a^{11} - \frac{501171}{2663593} a^{10} + \frac{1013558}{2663593} a^{9} + \frac{680637}{2663593} a^{8} - \frac{1035426}{2663593} a^{7} + \frac{1290760}{2663593} a^{6} - \frac{685238}{2663593} a^{5} + \frac{1163147}{2663593} a^{4} - \frac{394096}{2663593} a^{3} - \frac{978097}{2663593} a^{2} - \frac{1053511}{2663593} a + \frac{402138}{2663593}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{5069577}{2663593} a^{19} + \frac{15210989}{2663593} a^{18} - \frac{13772404}{2663593} a^{17} - \frac{5212947}{2663593} a^{16} + \frac{3481178}{2663593} a^{15} + \frac{38083464}{2663593} a^{14} - \frac{89138870}{2663593} a^{13} + \frac{95969693}{2663593} a^{12} - \frac{52519168}{2663593} a^{11} + \frac{16837722}{2663593} a^{10} - \frac{63630828}{2663593} a^{9} + \frac{132396579}{2663593} a^{8} - \frac{146687436}{2663593} a^{7} + \frac{116295184}{2663593} a^{6} - \frac{79084843}{2663593} a^{5} + \frac{20213325}{2663593} a^{4} + \frac{18815882}{2663593} a^{3} - \frac{27982575}{2663593} a^{2} + \frac{18220722}{2663593} a - \frac{3418100}{2663593} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 635.552521857 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 635.552521857 \cdot 1}{6\sqrt{3575480621700351753081}}\approx 0.169875858770$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 10.0.246071287.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$