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Properties

Label 20.4.70519039802...5929.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 43^{4}\cdot 67^{2}$
Root discriminant $22.00$
Ramified primes $11, 43, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -37, -191, -145, 272, 449, 218, -2346, 3045, -1773, 275, 489, -683, 548, -282, 57, 46, -51, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 25*x^18 - 51*x^17 + 46*x^16 + 57*x^15 - 282*x^14 + 548*x^13 - 683*x^12 + 489*x^11 + 275*x^10 - 1773*x^9 + 3045*x^8 - 2346*x^7 + 218*x^6 + 449*x^5 + 272*x^4 - 145*x^3 - 191*x^2 - 37*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 25*x^18 - 51*x^17 + 46*x^16 + 57*x^15 - 282*x^14 + 548*x^13 - 683*x^12 + 489*x^11 + 275*x^10 - 1773*x^9 + 3045*x^8 - 2346*x^7 + 218*x^6 + 449*x^5 + 272*x^4 - 145*x^3 - 191*x^2 - 37*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 25 x^{18} - 51 x^{17} + 46 x^{16} + 57 x^{15} - 282 x^{14} + 548 x^{13} - 683 x^{12} + 489 x^{11} + 275 x^{10} - 1773 x^{9} + 3045 x^{8} - 2346 x^{7} + 218 x^{6} + 449 x^{5} + 272 x^{4} - 145 x^{3} - 191 x^{2} - 37 x + 1 \)

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:  \(705190398026416240151735929=11^{16}\cdot 43^{4}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.00$
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:  $11, 43, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10582803619457100758699124307} a^{19} + \frac{2057415743511147149436359788}{10582803619457100758699124307} a^{18} + \frac{1786699364320985350663646329}{10582803619457100758699124307} a^{17} - \frac{3745077883608888911746961880}{10582803619457100758699124307} a^{16} + \frac{874358137956235160630213325}{10582803619457100758699124307} a^{15} - \frac{2967053117569958643303099853}{10582803619457100758699124307} a^{14} - \frac{1553402312914226583895992056}{10582803619457100758699124307} a^{13} + \frac{1286840151032443050505468697}{10582803619457100758699124307} a^{12} - \frac{5035891392046304630458393857}{10582803619457100758699124307} a^{11} + \frac{3241127731111296222852939926}{10582803619457100758699124307} a^{10} + \frac{2485650368851239678735080536}{10582803619457100758699124307} a^{9} - \frac{1032781071784749229222898461}{10582803619457100758699124307} a^{8} + \frac{1989673144947386924527436681}{10582803619457100758699124307} a^{7} - \frac{813592062247355969504087546}{10582803619457100758699124307} a^{6} + \frac{1593745128805503865392439846}{10582803619457100758699124307} a^{5} - \frac{628657136312436915347261478}{10582803619457100758699124307} a^{4} + \frac{3517079539253021050817034620}{10582803619457100758699124307} a^{3} - \frac{2488437822135619071472876916}{10582803619457100758699124307} a^{2} - \frac{3825760476051043514608838942}{10582803619457100758699124307} a - \frac{4701344542911336661971680172}{10582803619457100758699124307}$

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:  \( 272498.281701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 112 conjugacy class representatives for t20n263 are not computed
Character table for t20n263 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.617567936161.2, 10.10.617567936161.1, 10.2.396349570969.1

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 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
43Data not computed
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$