↑

Properties

Label 20.8.19522894749...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{12}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}$
Root discriminant $29.14$
Ramified primes $5, 97, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1040

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -278, -1555, -2959, -2114, 929, 2491, 1691, 1598, 848, -475, -339, -34, -127, 62, 40, -12, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 - 12*x^17 + 40*x^16 + 62*x^15 - 127*x^14 - 34*x^13 - 339*x^12 - 475*x^11 + 848*x^10 + 1598*x^9 + 1691*x^8 + 2491*x^7 + 929*x^6 - 2114*x^5 - 2959*x^4 - 1555*x^3 - 278*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 - 12*x^17 + 40*x^16 + 62*x^15 - 127*x^14 - 34*x^13 - 339*x^12 - 475*x^11 + 848*x^10 + 1598*x^9 + 1691*x^8 + 2491*x^7 + 929*x^6 - 2114*x^5 - 2959*x^4 - 1555*x^3 - 278*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} - 12 x^{17} + 40 x^{16} + 62 x^{15} - 127 x^{14} - 34 x^{13} - 339 x^{12} - 475 x^{11} + 848 x^{10} + 1598 x^{9} + 1691 x^{8} + 2491 x^{7} + 929 x^{6} - 2114 x^{5} - 2959 x^{4} - 1555 x^{3} - 278 x^{2} - 5 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(195228947498876925539306640625=5^{12}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.14$
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:  $5, 97, 419, 695771$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4296939631237616443124383466488997} a^{19} - \frac{1259481787188977038085882876515237}{4296939631237616443124383466488997} a^{18} + \frac{857284927357209442635537674114446}{4296939631237616443124383466488997} a^{17} + \frac{1713500138196172848096340276962247}{4296939631237616443124383466488997} a^{16} - \frac{958573371219409175057265274435172}{4296939631237616443124383466488997} a^{15} - \frac{1750615672927950264429194569575411}{4296939631237616443124383466488997} a^{14} + \frac{1770739367902147900993866076516876}{4296939631237616443124383466488997} a^{13} - \frac{460617399044955305147379778850232}{4296939631237616443124383466488997} a^{12} - \frac{350433962908515839740670674492656}{4296939631237616443124383466488997} a^{11} + \frac{611994094838236135887963803216460}{4296939631237616443124383466488997} a^{10} + \frac{682687461617282630168561690352205}{4296939631237616443124383466488997} a^{9} + \frac{1477536306158325321550463538454901}{4296939631237616443124383466488997} a^{8} - \frac{607761378621010583486188191685430}{4296939631237616443124383466488997} a^{7} - \frac{1232450399249030408808503908216189}{4296939631237616443124383466488997} a^{6} - \frac{381967211423779219100604820297483}{4296939631237616443124383466488997} a^{5} - \frac{103465209335145792041814999687123}{4296939631237616443124383466488997} a^{4} - \frac{1645065782818479886237017333140314}{4296939631237616443124383466488997} a^{3} + \frac{795775674395455895684478969892056}{4296939631237616443124383466488997} a^{2} - \frac{2114169299120572936240067750516966}{4296939631237616443124383466488997} a - \frac{502203014494980524657854981399821}{4296939631237616443124383466488997}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 11151713.5728 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1040:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1040 are not computed
Character table for t20n1040 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
97Data not computed
419Data not computed
695771Data not computed