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Properties

Label 20.0.88229434783...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $55.88$
Ramified primes $2, 5, 13$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1130272289, 12918360, 107996980, 108237080, 4225, -37056, 5616380, -520, -18190, -156800, -60920, 560, 5030, -360, 0, -8, 185, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 185*x^16 - 8*x^15 - 360*x^13 + 5030*x^12 + 560*x^11 - 60920*x^10 - 156800*x^9 - 18190*x^8 - 520*x^7 + 5616380*x^6 - 37056*x^5 + 4225*x^4 + 108237080*x^3 + 107996980*x^2 + 12918360*x + 1130272289)
 
gp: K = bnfinit(x^20 + 185*x^16 - 8*x^15 - 360*x^13 + 5030*x^12 + 560*x^11 - 60920*x^10 - 156800*x^9 - 18190*x^8 - 520*x^7 + 5616380*x^6 - 37056*x^5 + 4225*x^4 + 108237080*x^3 + 107996980*x^2 + 12918360*x + 1130272289, 1)
 

Normalized defining polynomial

\( x^{20} + 185 x^{16} - 8 x^{15} - 360 x^{13} + 5030 x^{12} + 560 x^{11} - 60920 x^{10} - 156800 x^{9} - 18190 x^{8} - 520 x^{7} + 5616380 x^{6} - 37056 x^{5} + 4225 x^{4} + 108237080 x^{3} + 107996980 x^{2} + 12918360 x + 1130272289 \)

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:  \(88229434783360000000000000000000000=2^{28}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.88$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12} a + \frac{5}{12}$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{16} - \frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{5}{12} a^{2} + \frac{5}{12}$, $\frac{1}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{19} + \frac{3457642530097266737140317553671139183472328322991584189774529187424447175577389}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{18} - \frac{2806563303895502425508446087536536621033063922905711962864535337853388975762693}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{17} - \frac{5298249702951789395513335859245536173670153566123716062687979653418941393596119}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{16} - \frac{1667789055996957432069960807558836772290823563272563200473418575751266983763477}{29661457867742331184463787667476150813682212154032140137344630950347067752429236} a^{15} + \frac{7899214141616256504132716710200582053668340266671391028273468251065852900051901}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{14} + \frac{20524435323029982915991616303414973803670755738021162674920130687681723140680105}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{13} + \frac{17292275579094589331454203002834520581386626415264955253768156469019274711284123}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{12} - \frac{1133184172655239814140267067491424071245839782355946524294386032917620554500021}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{11} + \frac{13510727754655956111567604499461049326881609390044702430614081121967108437802423}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{10} - \frac{496252844439426662330164716063368226864799753422349020691389025053424766115345}{9887152622580777061487929222492050271227404051344046712448210316782355917476412} a^{9} + \frac{13945379159490200043341108608194601300424360279485454391900061362264028895444161}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{8} + \frac{14371501810016840163627997313409951133420703355277753164687371043628002974781205}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{7} + \frac{608194817399522906734447326843614737779041770346536835597824405130205738711145}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{6} - \frac{16501801234168131956847760657086599288441638944102991148290350336609819805030841}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{5} - \frac{20201243793185826164758559762952892546519249424611374945233177169939092811989323}{88984373603226993553391363002428452441046636462096420412033892851041203257287708} a^{4} + \frac{18265002912978582544321536422279542485527604287437870921472104248740006709424237}{44492186801613496776695681501214226220523318231048210206016946425520601628643854} a^{3} - \frac{1777437690316305144166425322693531248074287744020313955214316074696778176631861}{14830728933871165592231893833738075406841106077016070068672315475173533876214618} a^{2} - \frac{7021504515136128668916712209968781385299056170624996728259445110768354389057271}{44492186801613496776695681501214226220523318231048210206016946425520601628643854} a + \frac{450823294690052990487576727394901540436087156256313883500861057888960355859953}{44492186801613496776695681501214226220523318231048210206016946425520601628643854}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 265229243.6396519 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-65}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-13})\), 5.1.50000.1, 10.0.297034400000000000.1, 10.0.59406880000000000.1, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$