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Properties

Label 20.0.15889137540...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{35}\cdot 11^{12}\cdot 61^{12}$
Root discriminant $1445.62$
Ramified primes $2, 5, 11, 61$
Class number Not computed
Class group Not computed
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![331470290341877498625616, 0, 0, 0, 0, -43581344807630608, 0, 0, 0, 0, 4673959352304, 0, 0, 0, 0, -1814102, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 1814102*x^15 + 4673959352304*x^10 - 43581344807630608*x^5 + 331470290341877498625616)
 
gp: K = bnfinit(x^20 - 1814102*x^15 + 4673959352304*x^10 - 43581344807630608*x^5 + 331470290341877498625616, 1)
 

Normalized defining polynomial

\( x^{20} - 1814102 x^{15} + 4673959352304 x^{10} - 43581344807630608 x^{5} + 331470290341877498625616 \)

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:  \(1588913754016079841241457850808227729797363281250000000000000000=2^{16}\cdot 5^{35}\cdot 11^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1445.62$
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:  $2, 5, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{122} a^{8} + \frac{19}{61} a^{3}$, $\frac{1}{122} a^{9} + \frac{19}{61} a^{4}$, $\frac{1}{163724} a^{10} - \frac{6569}{81862} a^{5}$, $\frac{1}{163724} a^{11} - \frac{6569}{81862} a^{6}$, $\frac{1}{9987164} a^{12} - \frac{907051}{4993582} a^{7} - \frac{21}{61} a^{2}$, $\frac{1}{109858804} a^{13} - \frac{6569}{54929402} a^{8} - \frac{152}{671} a^{3}$, $\frac{1}{6701387044} a^{14} - \frac{907051}{3350693522} a^{9} + \frac{18889}{40931} a^{4}$, $\frac{1}{12088067880330268460697160288312} a^{15} - \frac{6882306835106182441667015}{6044033940165134230348580144156} a^{10} - \frac{39496311979490506961219}{412469188858722059044222} a^{5} + \frac{162286270963407412}{1344125042644943569}$, $\frac{1}{12088067880330268460697160288312} a^{16} - \frac{6882306835106182441667015}{6044033940165134230348580144156} a^{11} - \frac{39496311979490506961219}{412469188858722059044222} a^{6} + \frac{162286270963407412}{1344125042644943569} a$, $\frac{1}{8111093547701610137127794553457352} a^{17} + \frac{66949677970605066127248723}{4055546773850805068563897276728676} a^{12} + \frac{65064438510728255831875079}{276766825724202501618672962} a^{7} + \frac{196404542497125168486}{901907903614757134799} a^{2}$, $\frac{1}{494776706409798218364795467760898472} a^{18} - \frac{302210246057951176717329967}{247388353204899109182397733880449236} a^{13} - \frac{13808189246030936765921546}{8441388184588176299369525341} a^{8} + \frac{23190351647024174803369}{55016382120500185222739} a^{3}$, $\frac{1}{331995170000974604522777758867562874712} a^{19} - \frac{4805961319206337339421189985}{165997585000487302261388879433781437356} a^{14} + \frac{16040686711775309496316708979}{5664171471858666296876951503811} a^{9} - \frac{12316139644216873552612586}{36915992402855624284457869} a^{4}$

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

Class group and class number

Not computed

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:  \( -\frac{5353954752695}{12088067880330268460697160288312} a^{15} + \frac{7029194475631031625}{6044033940165134230348580144156} a^{10} - \frac{911305950504919489}{412469188858722059044222} a^{5} + \frac{1172086258032368605}{1344125042644943569} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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

Sibling fields

Degree 25 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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$61$61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.4$x^{5} + 488$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$