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Properties

Label 20.8.26099879633...0256.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{2}$
Root discriminant $26.35$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 12, 28, -140, -467, 324, 1652, -764, -2662, 2160, 1408, -2768, 1123, 564, -800, 380, -57, -36, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 26*x^18 - 36*x^17 - 57*x^16 + 380*x^15 - 800*x^14 + 564*x^13 + 1123*x^12 - 2768*x^11 + 1408*x^10 + 2160*x^9 - 2662*x^8 - 764*x^7 + 1652*x^6 + 324*x^5 - 467*x^4 - 140*x^3 + 28*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 26*x^18 - 36*x^17 - 57*x^16 + 380*x^15 - 800*x^14 + 564*x^13 + 1123*x^12 - 2768*x^11 + 1408*x^10 + 2160*x^9 - 2662*x^8 - 764*x^7 + 1652*x^6 + 324*x^5 - 467*x^4 - 140*x^3 + 28*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 26 x^{18} - 36 x^{17} - 57 x^{16} + 380 x^{15} - 800 x^{14} + 564 x^{13} + 1123 x^{12} - 2768 x^{11} + 1408 x^{10} + 2160 x^{9} - 2662 x^{8} - 764 x^{7} + 1652 x^{6} + 324 x^{5} - 467 x^{4} - 140 x^{3} + 28 x^{2} + 12 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26099879633934179709805920256=2^{30}\cdot 11^{16}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.35$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{46} a^{17} + \frac{3}{46} a^{16} + \frac{5}{46} a^{15} - \frac{1}{23} a^{14} + \frac{5}{46} a^{13} + \frac{3}{23} a^{12} + \frac{1}{23} a^{11} + \frac{1}{46} a^{10} + \frac{9}{46} a^{9} - \frac{3}{46} a^{8} - \frac{5}{46} a^{7} - \frac{3}{23} a^{6} - \frac{3}{23} a^{5} - \frac{15}{46} a^{4} + \frac{11}{23} a^{3} - \frac{11}{23} a^{2} - \frac{5}{23} a - \frac{4}{23}$, $\frac{1}{46} a^{18} - \frac{2}{23} a^{16} + \frac{3}{23} a^{15} + \frac{11}{46} a^{14} - \frac{9}{46} a^{13} + \frac{7}{46} a^{12} - \frac{5}{46} a^{11} + \frac{3}{23} a^{10} - \frac{7}{46} a^{9} + \frac{2}{23} a^{8} - \frac{7}{23} a^{7} - \frac{11}{46} a^{6} + \frac{3}{46} a^{5} - \frac{1}{23} a^{4} - \frac{19}{46} a^{3} + \frac{5}{23} a^{2} + \frac{11}{23} a + \frac{1}{46}$, $\frac{1}{762722802963917767486} a^{19} - \frac{2777324767158480793}{381361401481958883743} a^{18} + \frac{5335251972862999009}{762722802963917767486} a^{17} + \frac{9156998745635225811}{762722802963917767486} a^{16} - \frac{132239245757804501}{762722802963917767486} a^{15} - \frac{29864788870136483223}{381361401481958883743} a^{14} - \frac{71026610278895972056}{381361401481958883743} a^{13} + \frac{94318038858074478183}{381361401481958883743} a^{12} - \frac{9360379460492176599}{381361401481958883743} a^{11} + \frac{42382381543301500126}{381361401481958883743} a^{10} - \frac{185762299050421252951}{381361401481958883743} a^{9} - \frac{8385321176167087256}{381361401481958883743} a^{8} - \frac{143484338395555075767}{762722802963917767486} a^{7} + \frac{330045857027613543257}{762722802963917767486} a^{6} + \frac{7028174563844720145}{381361401481958883743} a^{5} - \frac{348053464132580871767}{762722802963917767486} a^{4} - \frac{1461667260976957383}{33161860998431207282} a^{3} - \frac{191475796403268891447}{762722802963917767486} a^{2} - \frac{158220178716116719385}{762722802963917767486} a + \frac{127459286738692851203}{762722802963917767486}$

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:  \( 4908705.1386 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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

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
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$