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Properties

Label 20.12.9495293556...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{14}\cdot 11^{5}\cdot 9931^{5}$
Root discriminant $56.09$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 17500, 58425, -3950, -268435, -376080, -108306, 105143, 80809, 28001, 393, -11955, -4205, 769, 204, 104, 91, 2, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 15*x^18 + 2*x^17 + 91*x^16 + 104*x^15 + 204*x^14 + 769*x^13 - 4205*x^12 - 11955*x^11 + 393*x^10 + 28001*x^9 + 80809*x^8 + 105143*x^7 - 108306*x^6 - 376080*x^5 - 268435*x^4 - 3950*x^3 + 58425*x^2 + 17500*x + 125)
 
gp: K = bnfinit(x^20 - 2*x^19 - 15*x^18 + 2*x^17 + 91*x^16 + 104*x^15 + 204*x^14 + 769*x^13 - 4205*x^12 - 11955*x^11 + 393*x^10 + 28001*x^9 + 80809*x^8 + 105143*x^7 - 108306*x^6 - 376080*x^5 - 268435*x^4 - 3950*x^3 + 58425*x^2 + 17500*x + 125, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 15 x^{18} + 2 x^{17} + 91 x^{16} + 104 x^{15} + 204 x^{14} + 769 x^{13} - 4205 x^{12} - 11955 x^{11} + 393 x^{10} + 28001 x^{9} + 80809 x^{8} + 105143 x^{7} - 108306 x^{6} - 376080 x^{5} - 268435 x^{4} - 3950 x^{3} + 58425 x^{2} + 17500 x + 125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(94952935569355941136243902587890625=5^{14}\cdot 11^{5}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.09$
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:  $5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{18} - \frac{2}{25} a^{17} - \frac{3}{25} a^{15} - \frac{9}{25} a^{14} + \frac{9}{25} a^{13} - \frac{6}{25} a^{12} + \frac{4}{25} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{7}{25} a^{8} + \frac{1}{25} a^{7} + \frac{4}{25} a^{6} + \frac{8}{25} a^{5} + \frac{4}{25} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{90942578863452590111216357256372180058025} a^{19} + \frac{51435267925662947754554560900242695488}{90942578863452590111216357256372180058025} a^{18} - \frac{253411584515477877388010694755632908201}{3637703154538103604448654290254887202321} a^{17} + \frac{453358718705887598827062932080803736872}{90942578863452590111216357256372180058025} a^{16} - \frac{12140768590416061815438340492845658137149}{90942578863452590111216357256372180058025} a^{15} + \frac{15931997662883819137186864388081448056909}{90942578863452590111216357256372180058025} a^{14} + \frac{571751155783270175090070965041177551432}{1934948486456438087472688452263237873575} a^{13} + \frac{45185035432999924277426495590208527339344}{90942578863452590111216357256372180058025} a^{12} + \frac{120989527917088495052360515384664720652}{3637703154538103604448654290254887202321} a^{11} + \frac{6489430099227961164985213864980492815728}{18188515772690518022243271451274436011605} a^{10} + \frac{14045211875069840234930347129450525021458}{90942578863452590111216357256372180058025} a^{9} + \frac{27974899546859838233949824792396232439446}{90942578863452590111216357256372180058025} a^{8} + \frac{43198581000090046702726673403911852857609}{90942578863452590111216357256372180058025} a^{7} + \frac{25826646244060196659345391509946681922778}{90942578863452590111216357256372180058025} a^{6} + \frac{41561944889120138116411943325163686093004}{90942578863452590111216357256372180058025} a^{5} + \frac{2048334371226305714327285403364766302806}{18188515772690518022243271451274436011605} a^{4} + \frac{106932801425964640798923542155479955811}{3637703154538103604448654290254887202321} a^{3} + \frac{8144267250457134491271015262711862835908}{18188515772690518022243271451274436011605} a^{2} + \frac{52844681140954052841366727704969190062}{3637703154538103604448654290254887202321} a - \frac{882377209311056209965608160282170763232}{3637703154538103604448654290254887202321}$

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:  $15$
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:  \( 35948340235.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
9931Data not computed