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Properties

Label 20.0.64865899108...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $109.80$
Ramified primes $2, 5, 11, 13$
Class number $4842200$ (GRH)
Class group $[5, 968440]$ (GRH)
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![757112259541, -40054162896, 381913016562, -22085242938, 89758244598, -5511867932, 13002301543, -818308334, 1294117823, -80363506, 93261249, -5492826, 4979818, -266010, 196958, -8954, 5601, -192, 105, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 105*x^18 - 192*x^17 + 5601*x^16 - 8954*x^15 + 196958*x^14 - 266010*x^13 + 4979818*x^12 - 5492826*x^11 + 93261249*x^10 - 80363506*x^9 + 1294117823*x^8 - 818308334*x^7 + 13002301543*x^6 - 5511867932*x^5 + 89758244598*x^4 - 22085242938*x^3 + 381913016562*x^2 - 40054162896*x + 757112259541)
 
gp: K = bnfinit(x^20 - 2*x^19 + 105*x^18 - 192*x^17 + 5601*x^16 - 8954*x^15 + 196958*x^14 - 266010*x^13 + 4979818*x^12 - 5492826*x^11 + 93261249*x^10 - 80363506*x^9 + 1294117823*x^8 - 818308334*x^7 + 13002301543*x^6 - 5511867932*x^5 + 89758244598*x^4 - 22085242938*x^3 + 381913016562*x^2 - 40054162896*x + 757112259541, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 105 x^{18} - 192 x^{17} + 5601 x^{16} - 8954 x^{15} + 196958 x^{14} - 266010 x^{13} + 4979818 x^{12} - 5492826 x^{11} + 93261249 x^{10} - 80363506 x^{9} + 1294117823 x^{8} - 818308334 x^{7} + 13002301543 x^{6} - 5511867932 x^{5} + 89758244598 x^{4} - 22085242938 x^{3} + 381913016562 x^{2} - 40054162896 x + 757112259541 \)

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:  \(64865899108976789905836192655360000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.80$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2860=2^{2}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2860}(1,·)$, $\chi_{2860}(1871,·)$, $\chi_{2860}(1611,·)$, $\chi_{2860}(1351,·)$, $\chi_{2860}(521,·)$, $\chi_{2860}(779,·)$, $\chi_{2860}(1039,·)$, $\chi_{2860}(1299,·)$, $\chi_{2860}(1301,·)$, $\chi_{2860}(2391,·)$, $\chi_{2860}(729,·)$, $\chi_{2860}(1819,·)$, $\chi_{2860}(1769,·)$, $\chi_{2860}(2341,·)$, $\chi_{2860}(2599,·)$, $\chi_{2860}(2601,·)$, $\chi_{2860}(2029,·)$, $\chi_{2860}(2289,·)$, $\chi_{2860}(311,·)$, $\chi_{2860}(2809,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} - \frac{131}{331} a^{17} - \frac{161}{331} a^{16} - \frac{144}{331} a^{15} + \frac{60}{331} a^{14} + \frac{39}{331} a^{13} + \frac{119}{331} a^{12} - \frac{164}{331} a^{11} - \frac{139}{331} a^{10} + \frac{81}{331} a^{9} - \frac{88}{331} a^{8} - \frac{83}{331} a^{7} + \frac{123}{331} a^{6} - \frac{112}{331} a^{5} - \frac{37}{331} a^{4} - \frac{153}{331} a^{3} - \frac{73}{331} a^{2} + \frac{130}{331} a - \frac{8}{331}$, $\frac{1}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{19} + \frac{950043061168592580829701708067346021813753309439410773373936659422160137378537}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{18} - \frac{1215522733414123006974799311753707696679245947712471937703971161860596765908760021}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{17} - \frac{855452404093077796909945018066190099421204004893237726705729009294858353409631309}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{16} - \frac{2493514083329159864223700859525611450129081597834834548977209748762303653529106409}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{15} + \frac{1177719673850496716755751160335109537946840712355990038487781594263203500411198659}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{14} - \frac{1286850201045848472717169723555208997371892330153024670358099165014114501208153757}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{13} + \frac{870298318921828416628892048617599264607857082687550292453786549997780327106102788}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{12} + \frac{2347860678955912151872444606264258839452277411899239670808274951199685752565745415}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{11} - \frac{1168089838155776207461320002227863216314432429307886656055165610661175629936878704}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{10} + \frac{2250232403146359846808794034651119534982650627725940558601360244221812064170595260}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{9} - \frac{831699687338108018823683005394450465556024638928583718550748043854778047407498296}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{8} - \frac{156187938109694594556465905306593575148518645797898280532002006379166967315170613}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{7} - \frac{2688507043921891970076621297710670127917184623860750693735319179093938376900235940}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{6} - \frac{1608299519363974258117655920176635314135577482162093044769850519763717936760085891}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{5} - \frac{2259963820508046705236950718367638712167042780984815368704443818164526079548955381}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{4} + \frac{10619433477886266024801106970445969686818945513374632472195720508143568770235891}{127097462008011211243607559435317915687938659192624775934625443454861672326664209} a^{3} - \frac{1007177725456833740038664916340728533324185454257432447263321076563993484319307434}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a^{2} + \frac{1374665969608738533989101362594644021798842436906284732958118718049728593516195025}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987} a + \frac{1138233002921702215097909122037783333288704607694807579743415643871246874336126030}{5465190866344482083475125055718670374581362345282865365188894068559051910046560987}$

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

Class group and class number

$C_{5}\times C_{968440}$, which has order $4842200$ (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:  \( 140644.5991815038 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.254687846410025600000.1, 10.0.81500110851208192.3

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
13Data not computed