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Properties

Label 20.4.10175972532...5625.2
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 37^{15}$
Root discriminant $50.16$
Ramified primes $5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14125, -300750, -768525, -229360, 651681, 127193, -273260, -14126, 85072, 36628, -38460, -1392, -1384, 6652, -1716, -739, 296, 27, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 14*x^18 + 27*x^17 + 296*x^16 - 739*x^15 - 1716*x^14 + 6652*x^13 - 1384*x^12 - 1392*x^11 - 38460*x^10 + 36628*x^9 + 85072*x^8 - 14126*x^7 - 273260*x^6 + 127193*x^5 + 651681*x^4 - 229360*x^3 - 768525*x^2 - 300750*x - 14125)
 
gp: K = bnfinit(x^20 - 3*x^19 - 14*x^18 + 27*x^17 + 296*x^16 - 739*x^15 - 1716*x^14 + 6652*x^13 - 1384*x^12 - 1392*x^11 - 38460*x^10 + 36628*x^9 + 85072*x^8 - 14126*x^7 - 273260*x^6 + 127193*x^5 + 651681*x^4 - 229360*x^3 - 768525*x^2 - 300750*x - 14125, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 14 x^{18} + 27 x^{17} + 296 x^{16} - 739 x^{15} - 1716 x^{14} + 6652 x^{13} - 1384 x^{12} - 1392 x^{11} - 38460 x^{10} + 36628 x^{9} + 85072 x^{8} - 14126 x^{7} - 273260 x^{6} + 127193 x^{5} + 651681 x^{4} - 229360 x^{3} - 768525 x^{2} - 300750 x - 14125 \)

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:  \(10175972532709207369643951416015625=5^{15}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.16$
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:  $5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{35410} a^{18} - \frac{2723}{35410} a^{17} - \frac{1387}{17705} a^{16} + \frac{3966}{17705} a^{15} - \frac{3797}{17705} a^{14} - \frac{6212}{17705} a^{13} + \frac{1367}{17705} a^{12} + \frac{1956}{17705} a^{11} + \frac{2513}{17705} a^{10} + \frac{8364}{17705} a^{9} + \frac{151}{3541} a^{8} + \frac{2754}{17705} a^{7} + \frac{5251}{17705} a^{6} - \frac{8758}{17705} a^{5} + \frac{902}{3541} a^{4} - \frac{11927}{35410} a^{3} + \frac{8761}{35410} a^{2} + \frac{1266}{3541} a + \frac{444}{3541}$, $\frac{1}{17683284733345503062059607068831198035578533657993576900} a^{19} - \frac{62965613296993317811381310843581317293302992534939}{8841642366672751531029803534415599017789266828996788450} a^{18} + \frac{839548139091112809125627819991077313983816739129540443}{8841642366672751531029803534415599017789266828996788450} a^{17} - \frac{1283566983242645394060028984275795510752152012500072373}{17683284733345503062059607068831198035578533657993576900} a^{16} + \frac{460186297897823056511151298243728430837446267924553121}{17683284733345503062059607068831198035578533657993576900} a^{15} + \frac{1362654182021863771136230283579547811593753315602204009}{4420821183336375765514901767207799508894633414498394225} a^{14} - \frac{2013205043471562505340082503689952423465982040077609879}{4420821183336375765514901767207799508894633414498394225} a^{13} + \frac{1052137767084733791437156261353315110261861955437358463}{4420821183336375765514901767207799508894633414498394225} a^{12} + \frac{1832263679022674525336134621033588376608132311920519629}{4420821183336375765514901767207799508894633414498394225} a^{11} - \frac{561725740878736923962573985562048926254961021964223098}{4420821183336375765514901767207799508894633414498394225} a^{10} - \frac{65987260546789661883890189753496072253973997392617183}{884164236667275153102980353441559901778926682899678845} a^{9} - \frac{1694545099692942945851097438492939350431975532038475718}{4420821183336375765514901767207799508894633414498394225} a^{8} + \frac{374162783807668947839212365119156481501043578878471518}{4420821183336375765514901767207799508894633414498394225} a^{7} - \frac{2713480995546535296867911819657564899382057191948040713}{8841642366672751531029803534415599017789266828996788450} a^{6} - \frac{716213028177553229953119774761428136912462443255159801}{1768328473334550306205960706883119803557853365799357690} a^{5} - \frac{590711157965272646730995266784602695466472518358281657}{17683284733345503062059607068831198035578533657993576900} a^{4} - \frac{134790085672099681632018483763565694483553253981200161}{4420821183336375765514901767207799508894633414498394225} a^{3} - \frac{795176021767983719867882799287288072063153715478087821}{1768328473334550306205960706883119803557853365799357690} a^{2} + \frac{18573802483258195181742873373266853258940327729216199}{707331389333820122482384282753247921423141346319743076} a + \frac{109078930467453251482296518829308063297419486646469883}{707331389333820122482384282753247921423141346319743076}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1331606547.642767 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{185}) \), 4.4.6331625.1, 5.1.171125.1, 10.2.5417496640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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
5Data not computed
37Data not computed