↑

Properties

Label 20.0.12244191989...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}\cdot 29^{10}$
Root discriminant $90.03$
Ramified primes $5, 29$
Class number $639524$ (GRH)
Class group $[2, 319762]$ (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14282703401, -11221459620, 9783834900, -5735486785, 6398382025, -1816839501, 2338395640, -303606450, 511417970, -26505325, 67390996, -1239210, 5462275, -29505, 274400, -281, 8330, 0, 140, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 140*x^18 + 8330*x^16 - 281*x^15 + 274400*x^14 - 29505*x^13 + 5462275*x^12 - 1239210*x^11 + 67390996*x^10 - 26505325*x^9 + 511417970*x^8 - 303606450*x^7 + 2338395640*x^6 - 1816839501*x^5 + 6398382025*x^4 - 5735486785*x^3 + 9783834900*x^2 - 11221459620*x + 14282703401)
 
gp: K = bnfinit(x^20 + 140*x^18 + 8330*x^16 - 281*x^15 + 274400*x^14 - 29505*x^13 + 5462275*x^12 - 1239210*x^11 + 67390996*x^10 - 26505325*x^9 + 511417970*x^8 - 303606450*x^7 + 2338395640*x^6 - 1816839501*x^5 + 6398382025*x^4 - 5735486785*x^3 + 9783834900*x^2 - 11221459620*x + 14282703401, 1)
 

Normalized defining polynomial

\( x^{20} + 140 x^{18} + 8330 x^{16} - 281 x^{15} + 274400 x^{14} - 29505 x^{13} + 5462275 x^{12} - 1239210 x^{11} + 67390996 x^{10} - 26505325 x^{9} + 511417970 x^{8} - 303606450 x^{7} + 2338395640 x^{6} - 1816839501 x^{5} + 6398382025 x^{4} - 5735486785 x^{3} + 9783834900 x^{2} - 11221459620 x + 14282703401 \)

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:  \(1224419198989056167192757129669189453125=5^{35}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.03$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(725=5^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{725}(1,·)$, $\chi_{725}(581,·)$, $\chi_{725}(202,·)$, $\chi_{725}(204,·)$, $\chi_{725}(463,·)$, $\chi_{725}(146,·)$, $\chi_{725}(347,·)$, $\chi_{725}(28,·)$, $\chi_{725}(349,·)$, $\chi_{725}(608,·)$, $\chi_{725}(291,·)$, $\chi_{725}(492,·)$, $\chi_{725}(173,·)$, $\chi_{725}(494,·)$, $\chi_{725}(436,·)$, $\chi_{725}(57,·)$, $\chi_{725}(59,·)$, $\chi_{725}(637,·)$, $\chi_{725}(318,·)$, $\chi_{725}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{71} a^{10} - \frac{1}{71} a^{8} + \frac{11}{71} a^{6} - \frac{34}{71} a^{5} - \frac{32}{71} a^{4} + \frac{17}{71} a^{3} + \frac{30}{71} a^{2} - \frac{23}{71} a - \frac{20}{71}$, $\frac{1}{71} a^{11} - \frac{1}{71} a^{9} + \frac{11}{71} a^{7} - \frac{34}{71} a^{6} - \frac{32}{71} a^{5} + \frac{17}{71} a^{4} + \frac{30}{71} a^{3} - \frac{23}{71} a^{2} - \frac{20}{71} a$, $\frac{1}{71} a^{12} + \frac{10}{71} a^{8} - \frac{34}{71} a^{7} - \frac{21}{71} a^{6} - \frac{17}{71} a^{5} - \frac{2}{71} a^{4} - \frac{6}{71} a^{3} + \frac{10}{71} a^{2} - \frac{23}{71} a - \frac{20}{71}$, $\frac{1}{745401121921} a^{13} - \frac{3601962468}{745401121921} a^{12} + \frac{91}{745401121921} a^{11} + \frac{1894765867}{745401121921} a^{10} + \frac{3185}{745401121921} a^{9} + \frac{348396826963}{745401121921} a^{8} + \frac{220470807879}{745401121921} a^{7} - \frac{53838321263}{745401121921} a^{6} - \frac{293960568846}{745401121921} a^{5} + \frac{319253050135}{745401121921} a^{4} + \frac{251968105861}{745401121921} a^{3} + \frac{251775560125}{745401121921} a^{2} + \frac{83990388245}{745401121921} a - \frac{139472459499}{745401121921}$, $\frac{1}{745401121921} a^{14} + \frac{98}{745401121921} a^{12} + \frac{4216522574}{745401121921} a^{11} + \frac{3773}{745401121921} a^{10} + \frac{219686164688}{745401121921} a^{9} + \frac{72030}{745401121921} a^{8} + \frac{240496672651}{745401121921} a^{7} - \frac{83988152914}{745401121921} a^{6} + \frac{45843959261}{745401121921} a^{5} + \frac{41997723576}{745401121921} a^{4} - \frac{222220319788}{745401121921} a^{3} + \frac{293966770629}{745401121921} a^{2} + \frac{79439745754}{745401121921} a + \frac{1647086}{745401121921}$, $\frac{1}{745401121921} a^{15} + \frac{256194504}{745401121921} a^{12} - \frac{5145}{745401121921} a^{11} + \frac{2503287669}{745401121921} a^{10} - \frac{240100}{745401121921} a^{9} - \frac{170366950960}{745401121921} a^{8} + \frac{136477357673}{745401121921} a^{7} - \frac{200268023591}{745401121921} a^{6} + \frac{209932616956}{745401121921} a^{5} + \frac{28797249386}{745401121921} a^{4} + \frac{314814100505}{745401121921} a^{3} - \frac{38322552507}{745401121921} a^{2} - \frac{42142667144}{745401121921} a - \frac{179362065067}{745401121921}$, $\frac{1}{745401121921} a^{16} - \frac{5880}{745401121921} a^{12} + \frac{186802507}{745401121921} a^{11} - \frac{301840}{745401121921} a^{10} - \frac{261942538981}{745401121921} a^{9} - \frac{6482700}{745401121921} a^{8} + \frac{138413092015}{745401121921} a^{7} + \frac{52425270931}{745401121921} a^{6} + \frac{128046759681}{745401121921} a^{5} + \frac{346124625383}{745401121921} a^{4} - \frac{368109440614}{745401121921} a^{3} + \frac{188381981358}{745401121921} a^{2} + \frac{225753561396}{745401121921} a - \frac{172944030}{745401121921}$, $\frac{1}{745401121921} a^{17} - \frac{3661482366}{745401121921} a^{12} + \frac{233240}{745401121921} a^{11} + \frac{2723543343}{745401121921} a^{10} + \frac{12245100}{745401121921} a^{9} + \frac{108004626474}{745401121921} a^{8} + \frac{252213437640}{745401121921} a^{7} + \frac{291202905560}{745401121921} a^{6} - \frac{176236288007}{745401121921} a^{5} + \frac{3789214916}{745401121921} a^{4} + \frac{270865322375}{745401121921} a^{3} - \frac{194015584207}{745401121921} a^{2} - \frac{180154786788}{745401121921} a + \frac{336606804477}{745401121921}$, $\frac{1}{745401121921} a^{18} + \frac{279888}{745401121921} a^{12} - \frac{36996583}{745401121921} a^{11} + \frac{16163532}{745401121921} a^{10} + \frac{179363446680}{745401121921} a^{9} + \frac{370291824}{745401121921} a^{8} - \frac{324128265232}{745401121921} a^{7} + \frac{119516747389}{745401121921} a^{6} + \frac{311877187756}{745401121921} a^{5} + \frac{335118553170}{745401121921} a^{4} + \frac{264874487024}{745401121921} a^{3} + \frac{5548789173}{745401121921} a^{2} - \frac{212369429988}{745401121921} a + \frac{10976181104}{745401121921}$, $\frac{1}{745401121921} a^{19} - \frac{3733847476}{745401121921} a^{12} - \frac{9306276}{745401121921} a^{11} - \frac{5188740120}{745401121921} a^{10} - \frac{521151456}{745401121921} a^{9} + \frac{109149224413}{745401121921} a^{8} + \frac{94041892934}{745401121921} a^{7} - \frac{258609877709}{745401121921} a^{6} - \frac{112644292727}{745401121921} a^{5} + \frac{294646299635}{745401121921} a^{4} - \frac{34073801896}{745401121921} a^{3} - \frac{134617915311}{745401121921} a^{2} + \frac{181325737055}{745401121921} a - \frac{261974537784}{745401121921}$

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

Class group and class number

$C_{2}\times C_{319762}$, which has order $639524$ (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:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$29$29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$