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Properties

Label 20.0.10675123826...0000.9
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $282.76$
Ramified primes $2, 3, 5, 7$
Class number $342470656$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 22, 30404]$ (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![254263476601, -48089633940, 264017909460, 108843645740, 26808533785, 84561594444, 67604513890, -676066360, -6175508585, -416923380, 36950832, -10987120, 29832030, -102520, 1268390, -1644, 24405, -20, 240, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 240*x^18 - 20*x^17 + 24405*x^16 - 1644*x^15 + 1268390*x^14 - 102520*x^13 + 29832030*x^12 - 10987120*x^11 + 36950832*x^10 - 416923380*x^9 - 6175508585*x^8 - 676066360*x^7 + 67604513890*x^6 + 84561594444*x^5 + 26808533785*x^4 + 108843645740*x^3 + 264017909460*x^2 - 48089633940*x + 254263476601)
 
gp: K = bnfinit(x^20 + 240*x^18 - 20*x^17 + 24405*x^16 - 1644*x^15 + 1268390*x^14 - 102520*x^13 + 29832030*x^12 - 10987120*x^11 + 36950832*x^10 - 416923380*x^9 - 6175508585*x^8 - 676066360*x^7 + 67604513890*x^6 + 84561594444*x^5 + 26808533785*x^4 + 108843645740*x^3 + 264017909460*x^2 - 48089633940*x + 254263476601, 1)
 

Normalized defining polynomial

\( x^{20} + 240 x^{18} - 20 x^{17} + 24405 x^{16} - 1644 x^{15} + 1268390 x^{14} - 102520 x^{13} + 29832030 x^{12} - 10987120 x^{11} + 36950832 x^{10} - 416923380 x^{9} - 6175508585 x^{8} - 676066360 x^{7} + 67604513890 x^{6} + 84561594444 x^{5} + 26808533785 x^{4} + 108843645740 x^{3} + 264017909460 x^{2} - 48089633940 x + 254263476601 \)

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:  \(10675123826048640000000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.76$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4200=2^{3}\cdot 3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{4200}(1,·)$, $\chi_{4200}(1091,·)$, $\chi_{4200}(2759,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(1931,·)$, $\chi_{4200}(3599,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(2771,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(3611,·)$, $\chi_{4200}(349,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(1189,·)$, $\chi_{4200}(2029,·)$, $\chi_{4200}(239,·)$, $\chi_{4200}(2869,·)$, $\chi_{4200}(1079,·)$, $\chi_{4200}(251,·)$, $\chi_{4200}(3709,·)$, $\chi_{4200}(1919,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{175} a^{10} - \frac{1}{35} a^{8} + \frac{8}{35} a^{7} + \frac{3}{35} a^{6} - \frac{47}{175} a^{5} + \frac{9}{35} a^{4} + \frac{16}{35} a^{3} - \frac{6}{35} a^{2} + \frac{2}{5} a - \frac{51}{175}$, $\frac{1}{175} a^{11} - \frac{1}{35} a^{9} - \frac{2}{35} a^{8} - \frac{1}{5} a^{7} - \frac{47}{175} a^{6} - \frac{11}{35} a^{5} - \frac{2}{5} a^{4} + \frac{4}{35} a^{3} + \frac{2}{5} a^{2} - \frac{1}{175} a - \frac{1}{7}$, $\frac{1}{175} a^{12} - \frac{2}{35} a^{9} - \frac{2}{35} a^{8} + \frac{4}{25} a^{7} + \frac{4}{35} a^{6} - \frac{6}{35} a^{5} + \frac{9}{35} a^{4} + \frac{2}{5} a^{3} + \frac{24}{175} a^{2} - \frac{3}{7} a - \frac{11}{35}$, $\frac{1}{175} a^{13} - \frac{2}{35} a^{9} + \frac{3}{175} a^{8} - \frac{16}{35} a^{7} - \frac{11}{35} a^{6} - \frac{1}{7} a^{5} + \frac{2}{5} a^{4} - \frac{76}{175} a^{3} - \frac{1}{7} a^{2} - \frac{16}{35} a - \frac{12}{35}$, $\frac{1}{175} a^{14} + \frac{3}{175} a^{9} - \frac{1}{35} a^{8} - \frac{11}{35} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{7}{25} a^{4} - \frac{2}{7} a^{3} - \frac{6}{35} a^{2} - \frac{2}{35} a - \frac{2}{35}$, $\frac{1}{175} a^{15} - \frac{1}{35} a^{9} + \frac{2}{35} a^{8} + \frac{11}{35} a^{7} - \frac{4}{35} a^{6} - \frac{12}{35} a^{5} - \frac{1}{5} a^{4} + \frac{6}{35} a^{3} + \frac{16}{35} a^{2} + \frac{16}{35} a + \frac{3}{175}$, $\frac{1}{1225} a^{16} + \frac{2}{1225} a^{15} + \frac{1}{1225} a^{14} - \frac{3}{1225} a^{13} + \frac{3}{1225} a^{12} - \frac{3}{1225} a^{11} + \frac{2}{1225} a^{10} - \frac{32}{1225} a^{9} - \frac{24}{1225} a^{8} - \frac{306}{1225} a^{7} - \frac{304}{1225} a^{6} - \frac{284}{1225} a^{5} + \frac{509}{1225} a^{4} - \frac{22}{1225} a^{3} - \frac{363}{1225} a^{2} - \frac{239}{1225} a - \frac{271}{1225}$, $\frac{1}{368725} a^{17} + \frac{96}{368725} a^{16} + \frac{22}{52675} a^{15} - \frac{22}{10535} a^{14} - \frac{587}{368725} a^{13} + \frac{41}{368725} a^{12} - \frac{22}{10535} a^{11} - \frac{628}{368725} a^{10} + \frac{3679}{73745} a^{9} - \frac{244}{7525} a^{8} + \frac{105528}{368725} a^{7} + \frac{73}{73745} a^{6} - \frac{9557}{52675} a^{5} + \frac{358}{1505} a^{4} + \frac{35397}{368725} a^{3} - \frac{82598}{368725} a^{2} + \frac{36448}{368725} a - \frac{311}{1715}$, $\frac{1}{550011740909043193630099272767675} a^{18} + \frac{606082372109178327026660253}{550011740909043193630099272767675} a^{17} - \frac{27863988374534480509725374587}{78573105844149027661442753252525} a^{16} + \frac{35021123649200443216334223809}{15714621168829805532288550650505} a^{15} + \frac{263042958194790424225362910013}{110002348181808638726019854553535} a^{14} - \frac{93523637213035408861172252269}{550011740909043193630099272767675} a^{13} - \frac{127849888700202926286927293003}{78573105844149027661442753252525} a^{12} + \frac{261751312298334701911874537864}{110002348181808638726019854553535} a^{11} + \frac{1006227300333717156473891506911}{550011740909043193630099272767675} a^{10} - \frac{94360922949210838563000214278}{3142924233765961106457710130101} a^{9} + \frac{20058570304786845577374635187233}{550011740909043193630099272767675} a^{8} - \frac{259826971973528735168624414759673}{550011740909043193630099272767675} a^{7} - \frac{6857390164009056530239118951937}{15714621168829805532288550650505} a^{6} + \frac{3114427544966356387738118699769}{78573105844149027661442753252525} a^{5} - \frac{54713636293410681055563882413429}{110002348181808638726019854553535} a^{4} + \frac{225749045304891226966954307781302}{550011740909043193630099272767675} a^{3} - \frac{10964341106659742755863553120248}{22000469636361727745203970910707} a^{2} + \frac{196475489492013062384443215511713}{550011740909043193630099272767675} a - \frac{272107195890573853431326398691}{1827281531259279713056808215175}$, $\frac{1}{15008726153013382343626806003869496016492715833200346787541265175} a^{19} + \frac{519064176587912022622347525121}{15008726153013382343626806003869496016492715833200346787541265175} a^{18} + \frac{16267719508124841401961515615029682259387905679814573710579}{15008726153013382343626806003869496016492715833200346787541265175} a^{17} - \frac{829037352114314544076815277461979487565087350053341720321279}{3001745230602676468725361200773899203298543166640069357508253035} a^{16} + \frac{8484487607610398311322533388370328604724880761277727407143858}{15008726153013382343626806003869496016492715833200346787541265175} a^{15} + \frac{10297281672038320424660766642088396792078313861176580465025462}{15008726153013382343626806003869496016492715833200346787541265175} a^{14} + \frac{26596920350197071162388914426908828495353722497817870915446586}{15008726153013382343626806003869496016492715833200346787541265175} a^{13} - \frac{19884510066016981862008106467145866255998667145641310440459}{12252021349398679464185147758260813074687931292408446357176543} a^{12} + \frac{39182415842489559469300941458354796818624099651750051435916952}{15008726153013382343626806003869496016492715833200346787541265175} a^{11} + \frac{1134115064748170434941726499470162907553192148116972639273985}{600349046120535293745072240154779840659708633328013871501650607} a^{10} - \frac{442263571026964762008887249165858857233895036960186387556535534}{15008726153013382343626806003869496016492715833200346787541265175} a^{9} + \frac{457495991931081245486245492971887245714941754950725391537304523}{15008726153013382343626806003869496016492715833200346787541265175} a^{8} - \frac{369168608945097240587090951358802514623970051575120942734994461}{3001745230602676468725361200773899203298543166640069357508253035} a^{7} - \frac{2043572257799955685666705060950764543022626004718352518194444499}{15008726153013382343626806003869496016492715833200346787541265175} a^{6} + \frac{414510615220120321453529379092649244013047388416454849235419243}{3001745230602676468725361200773899203298543166640069357508253035} a^{5} + \frac{718023273696331073851375705580277968416528131913414681159836366}{15008726153013382343626806003869496016492715833200346787541265175} a^{4} - \frac{726124046026057397969319645749034903560813353161868037795863829}{2144103736144768906232400857695642288070387976171478112505895025} a^{3} + \frac{3648346065603576205002537975174724058402899488340673469623752362}{15008726153013382343626806003869496016492715833200346787541265175} a^{2} - \frac{984099076817899052421591408441068305812532356701903818346095677}{15008726153013382343626806003869496016492715833200346787541265175} a + \frac{40695956703031040856646318748992563309771139722115358712217783}{349040143093334473107600139624872000383551531004659227617238725}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{30404}$, which has order $342470656$ (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:  \( 180801817.57689384 \) (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

\(\Q(\sqrt{-42}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{15}, \sqrt{-42})\), 5.5.390625.1, 10.0.20420505000000000000000.1, 10.10.189843750000000000.1, 10.0.420175000000000000000.3

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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
5Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$