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Properties

Label 20.0.35754148095...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 601^{2}\cdot 1801^{2}$
Root discriminant $134.17$
Ramified primes $2, 5, 601, 1801$
Class number $41369600$ (GRH)
Class group $[2, 2, 2, 160, 32320]$ (GRH)
Galois group $C_4\times C_2^4:C_5$ (as 20T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5857959624005, 0, 18168127845025, 0, 7148125674525, 0, 1220914673775, 0, 113790309300, 0, 6306436255, 0, 214685475, 0, 4488850, 0, 55620, 0, 370, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005)
 
gp: K = bnfinit(x^20 + 370*x^18 + 55620*x^16 + 4488850*x^14 + 214685475*x^12 + 6306436255*x^10 + 113790309300*x^8 + 1220914673775*x^6 + 7148125674525*x^4 + 18168127845025*x^2 + 5857959624005, 1)
 

Normalized defining polynomial

\( x^{20} + 370 x^{18} + 55620 x^{16} + 4488850 x^{14} + 214685475 x^{12} + 6306436255 x^{10} + 113790309300 x^{8} + 1220914673775 x^{6} + 7148125674525 x^{4} + 18168127845025 x^{2} + 5857959624005 \)

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:  \(3575414809573364257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 601^{2}\cdot 1801^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.17$
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:  $2, 5, 601, 1801$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1043} a^{16} - \frac{20}{149} a^{14} + \frac{79}{1043} a^{12} + \frac{228}{1043} a^{10} + \frac{212}{1043} a^{8} + \frac{389}{1043} a^{6} - \frac{76}{1043} a^{4} - \frac{348}{1043} a^{2} + \frac{324}{1043}$, $\frac{1}{1043} a^{17} - \frac{20}{149} a^{15} + \frac{79}{1043} a^{13} + \frac{228}{1043} a^{11} + \frac{212}{1043} a^{9} + \frac{389}{1043} a^{7} - \frac{76}{1043} a^{5} - \frac{348}{1043} a^{3} + \frac{324}{1043} a$, $\frac{1}{13360087625866873550229342867571617687427791212949844093} a^{18} - \frac{6022074019250855824704674757590584749029401040414456}{13360087625866873550229342867571617687427791212949844093} a^{16} - \frac{679124690325100842637223513075808865828060536863328312}{13360087625866873550229342867571617687427791212949844093} a^{14} - \frac{613341664714669868752541948153286694873899778597961718}{1908583946552410507175620409653088241061113030421406299} a^{12} + \frac{5237220233850186394942994311266900953650914712050308247}{13360087625866873550229342867571617687427791212949844093} a^{10} + \frac{100488550832643005935036887787149708389047295436099730}{1908583946552410507175620409653088241061113030421406299} a^{8} - \frac{376786727421113701386087889805409826653992157389090902}{1908583946552410507175620409653088241061113030421406299} a^{6} - \frac{708893621774033301013206308964388024274868166863131577}{1908583946552410507175620409653088241061113030421406299} a^{4} - \frac{2577524350925574762253458366131217738306923629130285366}{13360087625866873550229342867571617687427791212949844093} a^{2} - \frac{5501437778997615091246706827539616160194541944550}{12343011163022644611589736952914509213708959260893}$, $\frac{1}{13360087625866873550229342867571617687427791212949844093} a^{19} - \frac{6022074019250855824704674757590584749029401040414456}{13360087625866873550229342867571617687427791212949844093} a^{17} - \frac{679124690325100842637223513075808865828060536863328312}{13360087625866873550229342867571617687427791212949844093} a^{15} - \frac{613341664714669868752541948153286694873899778597961718}{1908583946552410507175620409653088241061113030421406299} a^{13} + \frac{5237220233850186394942994311266900953650914712050308247}{13360087625866873550229342867571617687427791212949844093} a^{11} + \frac{100488550832643005935036887787149708389047295436099730}{1908583946552410507175620409653088241061113030421406299} a^{9} - \frac{376786727421113701386087889805409826653992157389090902}{1908583946552410507175620409653088241061113030421406299} a^{7} - \frac{708893621774033301013206308964388024274868166863131577}{1908583946552410507175620409653088241061113030421406299} a^{5} - \frac{2577524350925574762253458366131217738306923629130285366}{13360087625866873550229342867571617687427791212949844093} a^{3} - \frac{5501437778997615091246706827539616160194541944550}{12343011163022644611589736952914509213708959260893} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{160}\times C_{32320}$, which has order $41369600$ (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

$C_4\times C_2^4:C_5$ (as 20T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_4\times C_2^4:C_5$
Character table for $C_4\times C_2^4:C_5$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
601Data not computed
1801Data not computed