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Properties

Label 20.4.33067577086...2064.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{18}\cdot 241^{2}$
Root discriminant $21.18$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 31, -49, 111, -138, 354, -321, 520, -244, 297, -248, 258, -230, 140, -83, 58, -45, 23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 23*x^18 - 45*x^17 + 58*x^16 - 83*x^15 + 140*x^14 - 230*x^13 + 258*x^12 - 248*x^11 + 297*x^10 - 244*x^9 + 520*x^8 - 321*x^7 + 354*x^6 - 138*x^5 + 111*x^4 - 49*x^3 + 31*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 23*x^18 - 45*x^17 + 58*x^16 - 83*x^15 + 140*x^14 - 230*x^13 + 258*x^12 - 248*x^11 + 297*x^10 - 244*x^9 + 520*x^8 - 321*x^7 + 354*x^6 - 138*x^5 + 111*x^4 - 49*x^3 + 31*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 23 x^{18} - 45 x^{17} + 58 x^{16} - 83 x^{15} + 140 x^{14} - 230 x^{13} + 258 x^{12} - 248 x^{11} + 297 x^{10} - 244 x^{9} + 520 x^{8} - 321 x^{7} + 354 x^{6} - 138 x^{5} + 111 x^{4} - 49 x^{3} + 31 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(330675770864580911767512064=2^{10}\cdot 11^{18}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.18$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{2}{11} a^{9} + \frac{4}{11} a^{8} - \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{2}{11}$, $\frac{1}{11} a^{12} - \frac{2}{11} a$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{2}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{3}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{4}$, $\frac{1}{11} a^{16} - \frac{2}{11} a^{5}$, $\frac{1}{11} a^{17} - \frac{2}{11} a^{6}$, $\frac{1}{11} a^{18} - \frac{2}{11} a^{7}$, $\frac{1}{2285950481309} a^{19} + \frac{168020029}{207813680119} a^{18} - \frac{95409167654}{2285950481309} a^{17} + \frac{35120959398}{2285950481309} a^{16} - \frac{78146908779}{2285950481309} a^{15} + \frac{92956616388}{2285950481309} a^{14} - \frac{77425070810}{2285950481309} a^{13} + \frac{7795525613}{207813680119} a^{12} + \frac{1971663819}{207813680119} a^{11} + \frac{50567725304}{2285950481309} a^{10} - \frac{483660844827}{2285950481309} a^{9} + \frac{706627707651}{2285950481309} a^{8} + \frac{825040165739}{2285950481309} a^{7} - \frac{125512252203}{2285950481309} a^{6} - \frac{730060225369}{2285950481309} a^{5} - \frac{428208132754}{2285950481309} a^{4} - \frac{602728583980}{2285950481309} a^{3} + \frac{80075071786}{207813680119} a^{2} + \frac{525350340156}{2285950481309} a - \frac{85153168917}{2285950481309}$

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:  $11$
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:  \( 157711.328659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
241Data not computed