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Properties

Label 20.16.1140793109...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 5^{10}\cdot 61^{4}\cdot 3169^{2}\cdot 8951^{2}$
Root discriminant $56.61$
Ramified primes $2, 5, 61, 3169, 8951$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13845841, 0, -53794497, 0, 73638590, 0, -45402178, 0, 15118553, 0, -3056070, 0, 405861, 0, -36334, 0, 2104, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 70*x^18 + 2104*x^16 - 36334*x^14 + 405861*x^12 - 3056070*x^10 + 15118553*x^8 - 45402178*x^6 + 73638590*x^4 - 53794497*x^2 + 13845841)
 
gp: K = bnfinit(x^20 - 70*x^18 + 2104*x^16 - 36334*x^14 + 405861*x^12 - 3056070*x^10 + 15118553*x^8 - 45402178*x^6 + 73638590*x^4 - 53794497*x^2 + 13845841, 1)
 

Normalized defining polynomial

\( x^{20} - 70 x^{18} + 2104 x^{16} - 36334 x^{14} + 405861 x^{12} - 3056070 x^{10} + 15118553 x^{8} - 45402178 x^{6} + 73638590 x^{4} - 53794497 x^{2} + 13845841 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(114079310946033402667018240000000000=2^{20}\cdot 5^{10}\cdot 61^{4}\cdot 3169^{2}\cdot 8951^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.61$
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:  $2, 5, 61, 3169, 8951$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{61} a^{14} - \frac{9}{61} a^{12} + \frac{30}{61} a^{10} + \frac{22}{61} a^{8} + \frac{28}{61} a^{6} + \frac{30}{61} a^{4} + \frac{8}{61} a^{2}$, $\frac{1}{61} a^{15} - \frac{9}{61} a^{13} + \frac{30}{61} a^{11} + \frac{22}{61} a^{9} + \frac{28}{61} a^{7} + \frac{30}{61} a^{5} + \frac{8}{61} a^{3}$, $\frac{1}{3721} a^{16} - \frac{9}{3721} a^{14} + \frac{1555}{3721} a^{12} - \frac{1015}{3721} a^{10} + \frac{1614}{3721} a^{8} + \frac{579}{3721} a^{6} - \frac{1761}{3721} a^{4} - \frac{29}{61} a^{2}$, $\frac{1}{3721} a^{17} - \frac{9}{3721} a^{15} + \frac{1555}{3721} a^{13} - \frac{1015}{3721} a^{11} + \frac{1614}{3721} a^{9} + \frac{579}{3721} a^{7} - \frac{1761}{3721} a^{5} - \frac{29}{61} a^{3}$, $\frac{1}{6302044979011778469875779897} a^{18} - \frac{623703785121116134256314}{6302044979011778469875779897} a^{16} - \frac{46101566251879576687239220}{6302044979011778469875779897} a^{14} - \frac{2651460044103397684677812761}{6302044979011778469875779897} a^{12} - \frac{2756859837905889570862351115}{6302044979011778469875779897} a^{10} - \frac{855337973899565065470935408}{6302044979011778469875779897} a^{8} + \frac{2242523548485778863094718759}{6302044979011778469875779897} a^{6} + \frac{51327530836104193461799813}{103312212770684892948783277} a^{4} - \frac{149514161774486676109645}{1693642832306309720471857} a^{2} + \frac{5421753376254855692161}{27764636595185405253637}$, $\frac{1}{6302044979011778469875779897} a^{19} - \frac{623703785121116134256314}{6302044979011778469875779897} a^{17} - \frac{46101566251879576687239220}{6302044979011778469875779897} a^{15} - \frac{2651460044103397684677812761}{6302044979011778469875779897} a^{13} - \frac{2756859837905889570862351115}{6302044979011778469875779897} a^{11} - \frac{855337973899565065470935408}{6302044979011778469875779897} a^{9} + \frac{2242523548485778863094718759}{6302044979011778469875779897} a^{7} + \frac{51327530836104193461799813}{103312212770684892948783277} a^{5} - \frac{149514161774486676109645}{1693642832306309720471857} a^{3} + \frac{5421753376254855692161}{27764636595185405253637} a$

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:  $17$
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:  \( 43433793090.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.88642871875.1

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 32 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3169Data not computed
8951Data not computed