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Properties

Label 20.6.12671809176...9568.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 263$
Root discriminant $17.99$
Ramified primes $2, 11, 263$
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![1, -18, 41, 258, 315, 104, -69, -124, -66, 12, -11, -50, -39, -6, -2, 6, 3, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 3*x^16 + 6*x^15 - 2*x^14 - 6*x^13 - 39*x^12 - 50*x^11 - 11*x^10 + 12*x^9 - 66*x^8 - 124*x^7 - 69*x^6 + 104*x^5 + 315*x^4 + 258*x^3 + 41*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 4*x^17 + 3*x^16 + 6*x^15 - 2*x^14 - 6*x^13 - 39*x^12 - 50*x^11 - 11*x^10 + 12*x^9 - 66*x^8 - 124*x^7 - 69*x^6 + 104*x^5 + 315*x^4 + 258*x^3 + 41*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 4 x^{17} + 3 x^{16} + 6 x^{15} - 2 x^{14} - 6 x^{13} - 39 x^{12} - 50 x^{11} - 11 x^{10} + 12 x^{9} - 66 x^{8} - 124 x^{7} - 69 x^{6} + 104 x^{5} + 315 x^{4} + 258 x^{3} + 41 x^{2} - 18 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12671809176594786122989568=-\,2^{20}\cdot 11^{16}\cdot 263\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.99$
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:  $2, 11, 263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{883450351024569231793} a^{19} - \frac{158985213294494170741}{883450351024569231793} a^{18} - \frac{149736857671158173230}{883450351024569231793} a^{17} + \frac{256730315156123319817}{883450351024569231793} a^{16} + \frac{6646415978430652142}{883450351024569231793} a^{15} + \frac{217175177637082613140}{883450351024569231793} a^{14} + \frac{300517446072932130441}{883450351024569231793} a^{13} + \frac{423771135779010693073}{883450351024569231793} a^{12} + \frac{2093334490953772623}{38410884827155183991} a^{11} + \frac{293871519520888229279}{883450351024569231793} a^{10} + \frac{2916067262723304946}{883450351024569231793} a^{9} - \frac{4952627137255755322}{883450351024569231793} a^{8} - \frac{422289200081694042569}{883450351024569231793} a^{7} - \frac{226589130724908172152}{883450351024569231793} a^{6} + \frac{40373460089188401342}{883450351024569231793} a^{5} - \frac{397062709994354435689}{883450351024569231793} a^{4} + \frac{403833498640461333856}{883450351024569231793} a^{3} + \frac{76091929027676518515}{883450351024569231793} a^{2} - \frac{201312294611067825533}{883450351024569231793} a + \frac{251610912703859382277}{883450351024569231793}$

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:  $12$
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:  \( 54056.9333164 \) (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

\(\Q(\zeta_{11})^+\), 10.8.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $20$

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$2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$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}$
263Data not computed