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Properties

Label 20.0.13936805290...4336.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $50.96$
Ramified primes $2, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times A_4$ (as 20T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 0, 480, -1344, 3600, -3104, 6032, -1024, 7512, 848, 4144, 208, 1704, -216, 340, -100, 90, 4, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 4*x^17 + 90*x^16 - 100*x^15 + 340*x^14 - 216*x^13 + 1704*x^12 + 208*x^11 + 4144*x^10 + 848*x^9 + 7512*x^8 - 1024*x^7 + 6032*x^6 - 3104*x^5 + 3600*x^4 - 1344*x^3 + 480*x^2 + 32)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 4*x^17 + 90*x^16 - 100*x^15 + 340*x^14 - 216*x^13 + 1704*x^12 + 208*x^11 + 4144*x^10 + 848*x^9 + 7512*x^8 - 1024*x^7 + 6032*x^6 - 3104*x^5 + 3600*x^4 - 1344*x^3 + 480*x^2 + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 4 x^{17} + 90 x^{16} - 100 x^{15} + 340 x^{14} - 216 x^{13} + 1704 x^{12} + 208 x^{11} + 4144 x^{10} + 848 x^{9} + 7512 x^{8} - 1024 x^{7} + 6032 x^{6} - 3104 x^{5} + 3600 x^{4} - 1344 x^{3} + 480 x^{2} + 32 \)

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:  \(13936805290105078002151559671054336=2^{30}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.96$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{7791614083599745759671044816} a^{19} - \frac{99737463944323405425691521}{7791614083599745759671044816} a^{18} - \frac{86855204513627922866244967}{7791614083599745759671044816} a^{17} - \frac{177995961066533297114683313}{7791614083599745759671044816} a^{16} + \frac{8147361951811085738872671}{1947903520899936439917761204} a^{15} - \frac{137825642523301752103755983}{3895807041799872879835522408} a^{14} + \frac{120837199037024524332563479}{1947903520899936439917761204} a^{13} - \frac{20265821645264571626016395}{3895807041799872879835522408} a^{12} + \frac{223981195293517665853530275}{1947903520899936439917761204} a^{11} + \frac{736616161009700469544717}{1947903520899936439917761204} a^{10} + \frac{243240408124483629017308119}{1947903520899936439917761204} a^{9} - \frac{114038821616312436345888805}{973951760449968219958880602} a^{8} - \frac{28333275490980976250150751}{486975880224984109979440301} a^{7} + \frac{38266564595947952235364163}{973951760449968219958880602} a^{6} + \frac{31015203637653926577830502}{486975880224984109979440301} a^{5} + \frac{8160617050230749462557205}{486975880224984109979440301} a^{4} + \frac{176977979430592095399568924}{486975880224984109979440301} a^{3} + \frac{101963120894746782859185460}{486975880224984109979440301} a^{2} - \frac{59282949398974893714857022}{486975880224984109979440301} a - \frac{173507709961096364077655437}{486975880224984109979440301}$

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:  $9$
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:  \( 421620502.283 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times A_4$ (as 20T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 60
The 20 conjugacy class representatives for $C_5\times A_4$
Character table for $C_5\times A_4$

Intermediate fields

4.0.3136.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 sibling: 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 $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
7Data not computed
11Data not computed