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Properties

Label 20.8.20249698481...1853.1
Degree $20$
Signature $[8, 6]$
Discriminant $13^{13}\cdot 401^{8}$
Root discriminant $58.25$
Ramified primes $13, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9099, -56310, 97592, -81046, 208699, -261751, 98555, -33568, 29975, -14265, 16087, -3007, -4712, 270, 234, 442, -86, -18, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 - 18*x^17 - 86*x^16 + 442*x^15 + 234*x^14 + 270*x^13 - 4712*x^12 - 3007*x^11 + 16087*x^10 - 14265*x^9 + 29975*x^8 - 33568*x^7 + 98555*x^6 - 261751*x^5 + 208699*x^4 - 81046*x^3 + 97592*x^2 - 56310*x + 9099)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 - 18*x^17 - 86*x^16 + 442*x^15 + 234*x^14 + 270*x^13 - 4712*x^12 - 3007*x^11 + 16087*x^10 - 14265*x^9 + 29975*x^8 - 33568*x^7 + 98555*x^6 - 261751*x^5 + 208699*x^4 - 81046*x^3 + 97592*x^2 - 56310*x + 9099, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} - 18 x^{17} - 86 x^{16} + 442 x^{15} + 234 x^{14} + 270 x^{13} - 4712 x^{12} - 3007 x^{11} + 16087 x^{10} - 14265 x^{9} + 29975 x^{8} - 33568 x^{7} + 98555 x^{6} - 261751 x^{5} + 208699 x^{4} - 81046 x^{3} + 97592 x^{2} - 56310 x + 9099 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(202496984818189934420549080247241853=13^{13}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.25$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{1939811921331838530915484602680225185348233770577597} a^{19} - \frac{180216740524456809885737380116034038438061746382550}{1939811921331838530915484602680225185348233770577597} a^{18} - \frac{87084810556903654978382498053604362426627730792795}{646603973777279510305161534226741728449411256859199} a^{17} - \frac{652424822607002904416556346278837379324448478309943}{1939811921331838530915484602680225185348233770577597} a^{16} - \frac{948085392677095465131641671895535285176070124099439}{1939811921331838530915484602680225185348233770577597} a^{15} + \frac{709912886896975913212665609478812411678073413669083}{1939811921331838530915484602680225185348233770577597} a^{14} - \frac{15072431924218481894513990469595818121976105469277}{1939811921331838530915484602680225185348233770577597} a^{13} - \frac{150149980524812756246834199876363524629512171733370}{1939811921331838530915484602680225185348233770577597} a^{12} + \frac{724716134901879776517572814677263555610875590934811}{1939811921331838530915484602680225185348233770577597} a^{11} + \frac{296145623822617226979135099249115807070119930779826}{1939811921331838530915484602680225185348233770577597} a^{10} + \frac{148432610051568227292823675939695963316588003035218}{1939811921331838530915484602680225185348233770577597} a^{9} - \frac{352471441460925615003300091359258890832398041989124}{1939811921331838530915484602680225185348233770577597} a^{8} + \frac{201980493194160231770456409349632233461259501524990}{646603973777279510305161534226741728449411256859199} a^{7} - \frac{59586528695246260370137449840820859058492595549403}{646603973777279510305161534226741728449411256859199} a^{6} - \frac{218789030574659359839948351343300029206678906040181}{1939811921331838530915484602680225185348233770577597} a^{5} + \frac{346224143258799936942710552207169218465475901768037}{1939811921331838530915484602680225185348233770577597} a^{4} + \frac{560235364860188270085681710025483116839408127337209}{1939811921331838530915484602680225185348233770577597} a^{3} - \frac{110027981716302161987128711286508611792631564230287}{646603973777279510305161534226741728449411256859199} a^{2} + \frac{114327266907193509946946965997847090252571724459180}{646603973777279510305161534226741728449411256859199} a + \frac{210458907713064313430506914369328367442438787321215}{646603973777279510305161534226741728449411256859199}$

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:  $13$
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:  \( 19664730473.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.160801.1, 10.10.9600508843720093.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
401Data not computed