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Properties

Label 20.10.6586186841...0144.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, 75, -472, 492, 376, -1857, 2507, -1523, -250, 1548, -1683, 1042, -305, -159, 241, -119, 2, 30, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 10*x^18 + 30*x^17 + 2*x^16 - 119*x^15 + 241*x^14 - 159*x^13 - 305*x^12 + 1042*x^11 - 1683*x^10 + 1548*x^9 - 250*x^8 - 1523*x^7 + 2507*x^6 - 1857*x^5 + 376*x^4 + 492*x^3 - 472*x^2 + 75*x + 67)
 
gp: K = bnfinit(x^20 - 2*x^19 - 10*x^18 + 30*x^17 + 2*x^16 - 119*x^15 + 241*x^14 - 159*x^13 - 305*x^12 + 1042*x^11 - 1683*x^10 + 1548*x^9 - 250*x^8 - 1523*x^7 + 2507*x^6 - 1857*x^5 + 376*x^4 + 492*x^3 - 472*x^2 + 75*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 10 x^{18} + 30 x^{17} + 2 x^{16} - 119 x^{15} + 241 x^{14} - 159 x^{13} - 305 x^{12} + 1042 x^{11} - 1683 x^{10} + 1548 x^{9} - 250 x^{8} - 1523 x^{7} + 2507 x^{6} - 1857 x^{5} + 376 x^{4} + 492 x^{3} - 472 x^{2} + 75 x + 67 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-658618684118710741619590144=-\,2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{558115566026395600807} a^{19} - \frac{67974499839246942958}{558115566026395600807} a^{18} + \frac{174390056349978104251}{558115566026395600807} a^{17} - \frac{101555676929520562518}{558115566026395600807} a^{16} - \frac{104229577503767450298}{558115566026395600807} a^{15} + \frac{211484387583975452681}{558115566026395600807} a^{14} - \frac{7742427661144800057}{558115566026395600807} a^{13} - \frac{251779271992981764979}{558115566026395600807} a^{12} + \frac{122008902807154588176}{558115566026395600807} a^{11} + \frac{106442926712473175474}{558115566026395600807} a^{10} + \frac{229710878168922894533}{558115566026395600807} a^{9} - \frac{227435778961437068683}{558115566026395600807} a^{8} + \frac{48332323811693196256}{558115566026395600807} a^{7} - \frac{1376030036301654185}{6270961416026916863} a^{6} + \frac{208969463905112997908}{558115566026395600807} a^{5} + \frac{99963232012818994040}{558115566026395600807} a^{4} - \frac{136349663196818522457}{558115566026395600807} a^{3} + \frac{97114214822278612396}{558115566026395600807} a^{2} + \frac{207219164250025033294}{558115566026395600807} a + \frac{206775471898486014567}{558115566026395600807}$

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:  $14$
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:  \( 826263.9163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{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}$
241Data not computed