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Properties

Label 20.0.21137359210...3841.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{9}\cdot 19^{15}$
Root discriminant $46.37$
Ramified primes $3, 11, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11688877, 12555242, 13160320, 12840750, 8730182, 4025380, 1530472, 270476, -155820, -113524, 2312, 23760, 7629, 69, 379, 545, 170, -2, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 10*x^18 - 2*x^17 + 170*x^16 + 545*x^15 + 379*x^14 + 69*x^13 + 7629*x^12 + 23760*x^11 + 2312*x^10 - 113524*x^9 - 155820*x^8 + 270476*x^7 + 1530472*x^6 + 4025380*x^5 + 8730182*x^4 + 12840750*x^3 + 13160320*x^2 + 12555242*x + 11688877)
 
gp: K = bnfinit(x^20 - x^19 - 10*x^18 - 2*x^17 + 170*x^16 + 545*x^15 + 379*x^14 + 69*x^13 + 7629*x^12 + 23760*x^11 + 2312*x^10 - 113524*x^9 - 155820*x^8 + 270476*x^7 + 1530472*x^6 + 4025380*x^5 + 8730182*x^4 + 12840750*x^3 + 13160320*x^2 + 12555242*x + 11688877, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 10 x^{18} - 2 x^{17} + 170 x^{16} + 545 x^{15} + 379 x^{14} + 69 x^{13} + 7629 x^{12} + 23760 x^{11} + 2312 x^{10} - 113524 x^{9} - 155820 x^{8} + 270476 x^{7} + 1530472 x^{6} + 4025380 x^{5} + 8730182 x^{4} + 12840750 x^{3} + 13160320 x^{2} + 12555242 x + 11688877 \)

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:  \(2113735921053303828889591550033841=3^{10}\cdot 11^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.37$
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:  $3, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{1}{17} a^{17} - \frac{5}{17} a^{16} + \frac{8}{17} a^{15} + \frac{1}{17} a^{14} - \frac{7}{17} a^{13} - \frac{6}{17} a^{12} + \frac{2}{17} a^{11} - \frac{1}{17} a^{10} - \frac{2}{17} a^{9} - \frac{7}{17} a^{8} - \frac{1}{17} a^{7} - \frac{4}{17} a^{6} - \frac{5}{17} a^{5} + \frac{8}{17} a^{4} + \frac{2}{17} a^{3} - \frac{4}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{19} + \frac{25945541840197825493560422728560735502231064233574637622837235972893}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{18} - \frac{1353821782193476929511915875677394369215047583474775313459235380622224}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{17} - \frac{1515962478538228548610298021389047153934771781175900020235017430411628}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{16} + \frac{1303794091551578173607581320719470309932155161960176120756149288293822}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{15} + \frac{1225919850694886205479606975820404366018936155451089861024877683257544}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{14} - \frac{221290640243566218414807029765934322892576701288825274942869060894488}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{13} + \frac{961958653593717564595749974812609733618224221361087920241995572384740}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{12} + \frac{1624735902652143073110566796713538491587754439568326149474519135872423}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{11} + \frac{90228537358081628359043597650556375089076308313160413475083894353097}{198821814336006263284998839439673076385498270499707741198979751020841} a^{10} + \frac{193961282772894105679823296232308894602839803492502767538982548580278}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{9} - \frac{541808117391552509373917272073571977678691452703185301544642863932966}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{8} + \frac{431154417506263616694087931305422170461292389057489099890909418977249}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{7} - \frac{708040673362778901700136541769173986712172360802560609266778246576475}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{6} + \frac{1422935270888541461169725632510690763179372964157243788577417815878168}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{5} + \frac{417380823403146383603636310364619011020605160855260939924021919627076}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{4} - \frac{8313124193041079030029407677814509142870531781275344976738357201983}{31008906823046848402247525417196718335352941270596620186996841902333} a^{3} - \frac{1207344255257121129482857120484890436780083104418087953654344568340870}{3379970843712106475844980270474442298553470598495031600382655767354297} a^{2} - \frac{928553718416737822571244510168901320088296470972297782083860196968337}{3379970843712106475844980270474442298553470598495031600382655767354297} a - \frac{88552980131369503106146421828036481268602451436070646213641486961050}{198821814336006263284998839439673076385498270499707741198979751020841}$

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:  $9$
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:  \( 122053287.89335619 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.679041.1, 10.0.36252565459.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
3Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.10$x^{10} + 24057$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed