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Properties

Label 20.0.67955065915...000.15
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $438.14$
Ramified primes $2, 3, 5, 41$
Class number $1352567040$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 22, 960630]$ (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![1557775043258281, 0, 208980948821846, 0, 8674595358909, 0, -3055990120296, 0, 296029595250, 0, -14435792364, 0, 420958530, 0, -7498824, 0, 79845, 0, -442, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 442*x^18 + 79845*x^16 - 7498824*x^14 + 420958530*x^12 - 14435792364*x^10 + 296029595250*x^8 - 3055990120296*x^6 + 8674595358909*x^4 + 208980948821846*x^2 + 1557775043258281)
 
gp: K = bnfinit(x^20 - 442*x^18 + 79845*x^16 - 7498824*x^14 + 420958530*x^12 - 14435792364*x^10 + 296029595250*x^8 - 3055990120296*x^6 + 8674595358909*x^4 + 208980948821846*x^2 + 1557775043258281, 1)
 

Normalized defining polynomial

\( x^{20} - 442 x^{18} + 79845 x^{16} - 7498824 x^{14} + 420958530 x^{12} - 14435792364 x^{10} + 296029595250 x^{8} - 3055990120296 x^{6} + 8674595358909 x^{4} + 208980948821846 x^{2} + 1557775043258281 \)

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:  \(67955065915961530473358856164812748971048960000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.14$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(1091,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(1931,·)$, $\chi_{4920}(269,·)$, $\chi_{4920}(1229,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(851,·)$, $\chi_{4920}(2839,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(1759,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(4451,·)$, $\chi_{4920}(1829,·)$, $\chi_{4920}(2599,·)$, $\chi_{4920}(491,·)$, $\chi_{4920}(2669,·)$, $\chi_{4920}(1589,·)$, $\chi_{4920}(4159,·)$, $\chi_{4920}(3199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{4} - \frac{1}{6} a^{2} + \frac{1}{12}$, $\frac{1}{12} a^{5} - \frac{1}{6} a^{3} + \frac{1}{12} a$, $\frac{1}{72} a^{6} - \frac{1}{24} a^{4} - \frac{1}{8} a^{2} + \frac{11}{72}$, $\frac{1}{72} a^{7} - \frac{1}{24} a^{5} - \frac{1}{8} a^{3} + \frac{11}{72} a$, $\frac{1}{432} a^{8} + \frac{1}{216} a^{6} + \frac{1}{36} a^{4} - \frac{17}{216} a^{2} + \frac{19}{432}$, $\frac{1}{432} a^{9} + \frac{1}{216} a^{7} + \frac{1}{36} a^{5} - \frac{17}{216} a^{3} + \frac{19}{432} a$, $\frac{1}{864} a^{10} - \frac{1}{864} a^{8} - \frac{1}{144} a^{6} - \frac{17}{432} a^{4} - \frac{203}{864} a^{2} + \frac{9}{32}$, $\frac{1}{1728} a^{11} - \frac{1}{1728} a^{10} - \frac{1}{1728} a^{9} + \frac{1}{1728} a^{8} - \frac{1}{288} a^{7} + \frac{1}{288} a^{6} + \frac{19}{864} a^{5} - \frac{19}{864} a^{4} - \frac{347}{1728} a^{3} + \frac{347}{1728} a^{2} - \frac{61}{192} a + \frac{61}{192}$, $\frac{1}{15552} a^{12} - \frac{1}{5184} a^{8} - \frac{1}{1944} a^{6} - \frac{29}{1728} a^{4} - \frac{13}{648} a^{2} + \frac{5767}{15552}$, $\frac{1}{31104} a^{13} - \frac{1}{31104} a^{12} + \frac{11}{10368} a^{9} - \frac{11}{10368} a^{8} + \frac{1}{486} a^{7} - \frac{1}{486} a^{6} - \frac{125}{3456} a^{5} + \frac{125}{3456} a^{4} + \frac{11}{324} a^{3} - \frac{11}{324} a^{2} + \frac{5155}{31104} a - \frac{5155}{31104}$, $\frac{1}{31104} a^{14} - \frac{1}{31104} a^{12} - \frac{1}{10368} a^{10} - \frac{5}{31104} a^{8} + \frac{179}{31104} a^{6} + \frac{415}{10368} a^{4} - \frac{2993}{31104} a^{2} + \frac{1577}{31104}$, $\frac{1}{62208} a^{15} - \frac{1}{62208} a^{14} - \frac{1}{62208} a^{13} + \frac{1}{62208} a^{12} - \frac{1}{20736} a^{11} + \frac{1}{20736} a^{10} + \frac{67}{62208} a^{9} - \frac{67}{62208} a^{8} - \frac{109}{62208} a^{7} + \frac{109}{62208} a^{6} - \frac{593}{20736} a^{5} + \frac{593}{20736} a^{4} + \frac{8815}{62208} a^{3} - \frac{8815}{62208} a^{2} + \frac{24113}{62208} a - \frac{24113}{62208}$, $\frac{1}{122611968} a^{16} + \frac{281}{61305984} a^{14} + \frac{773}{61305984} a^{12} + \frac{34145}{61305984} a^{10} + \frac{18385}{30652992} a^{8} - \frac{191149}{61305984} a^{6} - \frac{2184277}{61305984} a^{4} + \frac{4467251}{61305984} a^{2} + \frac{16512091}{122611968}$, $\frac{1}{245223936} a^{17} - \frac{1}{245223936} a^{16} + \frac{281}{122611968} a^{15} - \frac{281}{122611968} a^{14} + \frac{773}{122611968} a^{13} - \frac{773}{122611968} a^{12} + \frac{34145}{122611968} a^{11} - \frac{34145}{122611968} a^{10} - \frac{52571}{61305984} a^{9} + \frac{52571}{61305984} a^{8} - \frac{474973}{122611968} a^{7} + \frac{474973}{122611968} a^{6} + \frac{1221611}{122611968} a^{5} - \frac{1221611}{122611968} a^{4} - \frac{925405}{122611968} a^{3} + \frac{925405}{122611968} a^{2} + \frac{21337099}{245223936} a - \frac{21337099}{245223936}$, $\frac{1}{41680783624572037022765621387297742844416} a^{18} - \frac{126564288941572138915546968302393}{41680783624572037022765621387297742844416} a^{16} - \frac{62124853848912802828836358630459633}{5210097953071504627845702673412217855552} a^{14} - \frac{97904001837341972214950233405484971}{5210097953071504627845702673412217855552} a^{12} - \frac{7022932113765937374410884950372607087}{20840391812286018511382810693648871422208} a^{10} + \frac{7550111267103327099251498797976505287}{20840391812286018511382810693648871422208} a^{8} + \frac{11086081286726474500370091314111612665}{2605048976535752313922851336706108927776} a^{6} + \frac{72020024180364049617160491596263265545}{2605048976535752313922851336706108927776} a^{4} + \frac{9024002005579519745078430708814820052949}{41680783624572037022765621387297742844416} a^{2} - \frac{1075328945502581535610526187453730685999}{13893594541524012340921873795765914281472}$, $\frac{1}{3290169271462035500373823094916723087591890316288} a^{19} - \frac{1}{83361567249144074045531242774595485688832} a^{18} - \frac{1849334676441900062291927464267885673933}{3290169271462035500373823094916723087591890316288} a^{17} + \frac{126564288941572138915546968302393}{83361567249144074045531242774595485688832} a^{16} + \frac{4325779869968328666507559946677334131973539}{822542317865508875093455773729180771897972579072} a^{15} - \frac{210761734757903858581377094558281985}{20840391812286018511382810693648871422208} a^{14} - \frac{13116162279136579015928118326087601285966725}{822542317865508875093455773729180771897972579072} a^{13} - \frac{139203438781045519809149345008231309}{20840391812286018511382810693648871422208} a^{12} + \frac{72343630890550622192843221521704520313513571}{1645084635731017750186911547458361543795945158144} a^{11} + \frac{9033000768500314159845183821287814593}{41680783624572037022765621387297742844416} a^{10} - \frac{557461627464376109635280995112769146114937523}{1645084635731017750186911547458361543795945158144} a^{9} + \frac{48061788180547763964430769964010902379}{41680783624572037022765621387297742844416} a^{8} + \frac{4418713365161888901537629544606925995502466035}{822542317865508875093455773729180771897972579072} a^{7} - \frac{50709542553564757822022311681011274429}{20840391812286018511382810693648871422208} a^{6} - \frac{29984442552924430664395210590193877637466655441}{822542317865508875093455773729180771897972579072} a^{5} - \frac{240843483335198344010935942918545685789}{20840391812286018511382810693648871422208} a^{4} - \frac{781099242828992362997187556414950992528849529887}{3290169271462035500373823094916723087591890316288} a^{3} - \frac{7457488500656528770296633788748235003273}{83361567249144074045531242774595485688832} a^{2} - \frac{41998615192446967703145205500913527441974146087}{1096723090487345166791274364972241029197296772096} a + \frac{9723537662016419078453517396295804580147}{27787189083048024681843747591531828562944}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{960630}$, which has order $1352567040$ (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:  \( 34503946347.31743 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{246}) \), \(\Q(\sqrt{-1230}) \), \(\Q(\sqrt{-5}, \sqrt{246})\), 5.5.2825761.1, 10.0.25551760733187200000.2, 10.10.2606819247971779313664.1, 10.0.8146310149911810355200000.1

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 R R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.10.9.8$x^{10} + 318816$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.8$x^{10} + 318816$$10$$1$$9$$C_{10}$$[\ ]_{10}$