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Properties

Label 20.0.34371041692...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{34}$
Root discriminant $26.72$
Ramified primes $3, 5$
Class number $11$
Class group $[11]$
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 50, -115, 600, -374, 1200, -375, 1620, -250, 1252, -75, 700, -15, 250, -1, 65, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 + 65*x^16 - x^15 + 250*x^14 - 15*x^13 + 700*x^12 - 75*x^11 + 1252*x^10 - 250*x^9 + 1620*x^8 - 375*x^7 + 1200*x^6 - 374*x^5 + 600*x^4 - 115*x^3 + 50*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^20 + 10*x^18 + 65*x^16 - x^15 + 250*x^14 - 15*x^13 + 700*x^12 - 75*x^11 + 1252*x^10 - 250*x^9 + 1620*x^8 - 375*x^7 + 1200*x^6 - 374*x^5 + 600*x^4 - 115*x^3 + 50*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} + 65 x^{16} - x^{15} + 250 x^{14} - 15 x^{13} + 700 x^{12} - 75 x^{11} + 1252 x^{10} - 250 x^{9} + 1620 x^{8} - 375 x^{7} + 1200 x^{6} - 374 x^{5} + 600 x^{4} - 115 x^{3} + 50 x^{2} + 5 x + 1 \)

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:  \(34371041692793369293212890625=3^{10}\cdot 5^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.72$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(75=3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{75}(64,·)$, $\chi_{75}(1,·)$, $\chi_{75}(4,·)$, $\chi_{75}(71,·)$, $\chi_{75}(74,·)$, $\chi_{75}(11,·)$, $\chi_{75}(14,·)$, $\chi_{75}(16,·)$, $\chi_{75}(19,·)$, $\chi_{75}(26,·)$, $\chi_{75}(29,·)$, $\chi_{75}(31,·)$, $\chi_{75}(34,·)$, $\chi_{75}(41,·)$, $\chi_{75}(44,·)$, $\chi_{75}(46,·)$, $\chi_{75}(49,·)$, $\chi_{75}(56,·)$, $\chi_{75}(59,·)$, $\chi_{75}(61,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{17} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{123895888665357296201} a^{19} + \frac{1074778453821408592}{123895888665357296201} a^{18} - \frac{17278477562052529184}{123895888665357296201} a^{17} + \frac{9821455351051768302}{123895888665357296201} a^{16} + \frac{40352716861929356910}{123895888665357296201} a^{15} - \frac{17443055563607815565}{123895888665357296201} a^{14} + \frac{17256796252429575151}{123895888665357296201} a^{13} - \frac{49146316383663203989}{123895888665357296201} a^{12} + \frac{34753366905137198463}{123895888665357296201} a^{11} + \frac{24225282215666387505}{123895888665357296201} a^{10} + \frac{5082425595277210562}{123895888665357296201} a^{9} + \frac{3331691963215954692}{123895888665357296201} a^{8} + \frac{3636578360876401158}{17699412666479613743} a^{7} - \frac{33057243118958634105}{123895888665357296201} a^{6} - \frac{9182417943214236376}{123895888665357296201} a^{5} + \frac{48916143508119544365}{123895888665357296201} a^{4} - \frac{9618844111753069192}{123895888665357296201} a^{3} + \frac{24946186641753538699}{123895888665357296201} a^{2} + \frac{19334146281372809205}{123895888665357296201} a + \frac{60197056974878048053}{123895888665357296201}$

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

Class group and class number

$C_{11}$, which has order $11$

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:  \( -\frac{13774208472919240275}{123895888665357296201} a^{19} + \frac{382066827692298620}{17699412666479613743} a^{18} - \frac{137164077990950747325}{123895888665357296201} a^{17} + \frac{26470511705567367715}{123895888665357296201} a^{16} - \frac{889559607599424633212}{123895888665357296201} a^{15} + \frac{185126722381440625300}{123895888665357296201} a^{14} - \frac{3408847727467512056245}{123895888665357296201} a^{13} + \frac{859144571943349812725}{123895888665357296201} a^{12} - \frac{1362653781943486195795}{17699412666479613743} a^{11} + \frac{2842604995199868420799}{123895888665357296201} a^{10} - \frac{17042908231073900963350}{123895888665357296201} a^{9} + \frac{6608762265921429897110}{123895888665357296201} a^{8} - \frac{22257800152921173475725}{123895888665357296201} a^{7} + \frac{9151778838286061491945}{123895888665357296201} a^{6} - \frac{16573754873024478540924}{123895888665357296201} a^{5} + \frac{1132183280788913512100}{17699412666479613743} a^{4} - \frac{8555215306098145906225}{123895888665357296201} a^{3} + \frac{2961654402315400227050}{123895888665357296201} a^{2} - \frac{634381023420555017655}{123895888665357296201} a + \frac{63097548086625515913}{123895888665357296201} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 161406.837641 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.185394287109375.1, 10.0.37078857421875.1

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

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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed