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Properties

Label 20.4.67729133686...9952.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 11^{16}\cdot 14057$
Root discriminant $21.96$
Ramified primes $2, 11, 14057$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 44, -77, 22, 242, -648, 884, -706, 205, 130, -225, 306, -397, 344, -202, 66, -9, 6, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 + 6*x^17 - 9*x^16 + 66*x^15 - 202*x^14 + 344*x^13 - 397*x^12 + 306*x^11 - 225*x^10 + 130*x^9 + 205*x^8 - 706*x^7 + 884*x^6 - 648*x^5 + 242*x^4 + 22*x^3 - 77*x^2 + 44*x - 11)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 + 6*x^17 - 9*x^16 + 66*x^15 - 202*x^14 + 344*x^13 - 397*x^12 + 306*x^11 - 225*x^10 + 130*x^9 + 205*x^8 - 706*x^7 + 884*x^6 - 648*x^5 + 242*x^4 + 22*x^3 - 77*x^2 + 44*x - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} + 6 x^{17} - 9 x^{16} + 66 x^{15} - 202 x^{14} + 344 x^{13} - 397 x^{12} + 306 x^{11} - 225 x^{10} + 130 x^{9} + 205 x^{8} - 706 x^{7} + 884 x^{6} - 648 x^{5} + 242 x^{4} + 22 x^{3} - 77 x^{2} + 44 x - 11 \)

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:  \(677291336864611819508989952=2^{20}\cdot 11^{16}\cdot 14057\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.96$
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:  $2, 11, 14057$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{381912990957865991095186991} a^{19} - \frac{146774862153429460919707253}{381912990957865991095186991} a^{18} + \frac{113916059830518228741065385}{381912990957865991095186991} a^{17} + \frac{26133347311286758105562180}{381912990957865991095186991} a^{16} - \frac{9759215702375079366053033}{381912990957865991095186991} a^{15} + \frac{23214976684273086797808397}{381912990957865991095186991} a^{14} + \frac{154010773148588576486862203}{381912990957865991095186991} a^{13} - \frac{170884567504439302764280188}{381912990957865991095186991} a^{12} + \frac{175488449818721973246948003}{381912990957865991095186991} a^{11} + \frac{124909627673231414566049777}{381912990957865991095186991} a^{10} + \frac{126285790446585728095318926}{381912990957865991095186991} a^{9} - \frac{25003327253520309811031546}{381912990957865991095186991} a^{8} - \frac{147228175653955844498849043}{381912990957865991095186991} a^{7} + \frac{188966169839569678063961946}{381912990957865991095186991} a^{6} - \frac{55231804526906875428137174}{381912990957865991095186991} a^{5} - \frac{6282019660599934119177402}{16604912650341999612834217} a^{4} + \frac{5625404089701092730666298}{381912990957865991095186991} a^{3} + \frac{148350603536053812837209324}{381912990957865991095186991} a^{2} - \frac{154034300699348745891415333}{381912990957865991095186991} a + \frac{78521481482919871220466747}{381912990957865991095186991}$

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:  \( 348570.600274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

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 t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
14057Data not computed