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Properties

Label 20.20.7328106503...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{16}\cdot 6029^{6}$
Root discriminant $49.35$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![379, -2171, -2826, 22227, 19449, -76144, -68101, 120651, 117415, -94221, -104150, 33124, 46783, -3292, -9971, -257, 1043, 54, -52, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 52*x^18 + 54*x^17 + 1043*x^16 - 257*x^15 - 9971*x^14 - 3292*x^13 + 46783*x^12 + 33124*x^11 - 104150*x^10 - 94221*x^9 + 117415*x^8 + 120651*x^7 - 68101*x^6 - 76144*x^5 + 19449*x^4 + 22227*x^3 - 2826*x^2 - 2171*x + 379)
 
gp: K = bnfinit(x^20 - 2*x^19 - 52*x^18 + 54*x^17 + 1043*x^16 - 257*x^15 - 9971*x^14 - 3292*x^13 + 46783*x^12 + 33124*x^11 - 104150*x^10 - 94221*x^9 + 117415*x^8 + 120651*x^7 - 68101*x^6 - 76144*x^5 + 19449*x^4 + 22227*x^3 - 2826*x^2 - 2171*x + 379, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 52 x^{18} + 54 x^{17} + 1043 x^{16} - 257 x^{15} - 9971 x^{14} - 3292 x^{13} + 46783 x^{12} + 33124 x^{11} - 104150 x^{10} - 94221 x^{9} + 117415 x^{8} + 120651 x^{7} - 68101 x^{6} - 76144 x^{5} + 19449 x^{4} + 22227 x^{3} - 2826 x^{2} - 2171 x + 379 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7328106503776662626239166259765625=5^{16}\cdot 6029^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.35$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4787039922414241497209802892763} a^{19} + \frac{318161010182054906029506479168}{683862846059177356744257556109} a^{18} + \frac{395605263524171599482467856354}{4787039922414241497209802892763} a^{17} + \frac{1415002135470690709591173208438}{4787039922414241497209802892763} a^{16} + \frac{1646847794811715368020315436813}{4787039922414241497209802892763} a^{15} + \frac{1475622254113385434618913409840}{4787039922414241497209802892763} a^{14} - \frac{40486677737433748284683057841}{683862846059177356744257556109} a^{13} - \frac{1653050903239184914500139781245}{4787039922414241497209802892763} a^{12} - \frac{934041260708850240206881357718}{4787039922414241497209802892763} a^{11} + \frac{1928846648947695775325784673689}{4787039922414241497209802892763} a^{10} - \frac{2171566534498565265315244172159}{4787039922414241497209802892763} a^{9} + \frac{1982332154499589463287365676863}{4787039922414241497209802892763} a^{8} + \frac{1630949550319520050096444049435}{4787039922414241497209802892763} a^{7} - \frac{2304514582046863333519379414360}{4787039922414241497209802892763} a^{6} + \frac{2130055328111871822500549523153}{4787039922414241497209802892763} a^{5} + \frac{475096881392840390304991833237}{4787039922414241497209802892763} a^{4} + \frac{2074568443163360658441919962165}{4787039922414241497209802892763} a^{3} - \frac{83007350090575886712540727841}{683862846059177356744257556109} a^{2} - \frac{1784439515873728046143768753934}{4787039922414241497209802892763} a + \frac{352851258686527642510179801821}{4787039922414241497209802892763}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 30263639022.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1, 10.10.85604360308203125.1, 10.10.17120872061640625.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed