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Properties

Label 20.0.87354219101...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 17^{15}$
Root discriminant $27.99$
Ramified primes $5, 17$
Class number $4$
Class group $[4]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85, -850, 4080, -12495, 27366, -45528, 59806, -64072, 57805, -45048, 30458, -17511, 8298, -3114, 843, -119, -6, -1, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 - x^17 - 6*x^16 - 119*x^15 + 843*x^14 - 3114*x^13 + 8298*x^12 - 17511*x^11 + 30458*x^10 - 45048*x^9 + 57805*x^8 - 64072*x^7 + 59806*x^6 - 45528*x^5 + 27366*x^4 - 12495*x^3 + 4080*x^2 - 850*x + 85)
 
gp: K = bnfinit(x^20 - 4*x^19 + 7*x^18 - x^17 - 6*x^16 - 119*x^15 + 843*x^14 - 3114*x^13 + 8298*x^12 - 17511*x^11 + 30458*x^10 - 45048*x^9 + 57805*x^8 - 64072*x^7 + 59806*x^6 - 45528*x^5 + 27366*x^4 - 12495*x^3 + 4080*x^2 - 850*x + 85, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 7 x^{18} - x^{17} - 6 x^{16} - 119 x^{15} + 843 x^{14} - 3114 x^{13} + 8298 x^{12} - 17511 x^{11} + 30458 x^{10} - 45048 x^{9} + 57805 x^{8} - 64072 x^{7} + 59806 x^{6} - 45528 x^{5} + 27366 x^{4} - 12495 x^{3} + 4080 x^{2} - 850 x + 85 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87354219101251702667236328125=5^{15}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.99$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{80} a^{16} + \frac{9}{80} a^{15} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} - \frac{1}{20} a^{12} + \frac{1}{8} a^{11} - \frac{11}{80} a^{10} - \frac{1}{8} a^{9} + \frac{13}{40} a^{8} + \frac{7}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{10} a^{5} - \frac{19}{80} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{7}{16}$, $\frac{1}{400} a^{17} + \frac{1}{200} a^{16} - \frac{33}{400} a^{15} + \frac{3}{80} a^{14} + \frac{1}{400} a^{13} - \frac{1}{200} a^{12} + \frac{99}{400} a^{11} - \frac{173}{400} a^{10} - \frac{23}{50} a^{9} - \frac{87}{400} a^{8} - \frac{1}{16} a^{7} + \frac{41}{100} a^{6} + \frac{113}{400} a^{5} + \frac{3}{400} a^{4} - \frac{19}{40} a^{2} + \frac{29}{80} a - \frac{39}{80}$, $\frac{1}{4082800} a^{18} - \frac{105}{163312} a^{17} - \frac{1743}{1020700} a^{16} - \frac{191459}{4082800} a^{15} + \frac{247073}{2041400} a^{14} - \frac{218377}{2041400} a^{13} - \frac{93607}{4082800} a^{12} - \frac{399}{1020700} a^{11} - \frac{822639}{2041400} a^{10} + \frac{1331431}{4082800} a^{9} + \frac{332107}{2041400} a^{8} + \frac{488941}{1020700} a^{7} + \frac{93017}{816560} a^{6} - \frac{122297}{1020700} a^{5} + \frac{170617}{2041400} a^{4} + \frac{1529}{102070} a^{3} - \frac{66811}{163312} a^{2} + \frac{157749}{408280} a - \frac{51293}{204140}$, $\frac{1}{396031600} a^{19} - \frac{1}{198015800} a^{18} - \frac{359867}{396031600} a^{17} - \frac{158643}{39603160} a^{16} + \frac{15321637}{198015800} a^{15} - \frac{355247}{3356200} a^{14} - \frac{1232134}{24751975} a^{13} + \frac{4742883}{396031600} a^{12} - \frac{23887779}{99007900} a^{11} + \frac{20827761}{198015800} a^{10} - \frac{189779943}{396031600} a^{9} - \frac{26406381}{99007900} a^{8} - \frac{48065317}{99007900} a^{7} + \frac{161575287}{396031600} a^{6} - \frac{2788215}{7920632} a^{5} - \frac{48059893}{396031600} a^{4} + \frac{3477091}{79206320} a^{3} - \frac{18828977}{79206320} a^{2} - \frac{3487049}{39603160} a + \frac{14798069}{79206320}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1365722.71668 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{85}) \), 4.0.614125.1, 5.1.614125.1 x5, 10.2.32057708828125.2 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.614125.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$