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Properties

Label 20.20.7588612227...6777.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 2777^{5}$
Root discriminant $49.43$
Ramified primes $11, 2777$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times S_4$ (as 20T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 296, -785, -2522, 8990, 7629, -32303, -11604, 52063, 10050, -42734, -5250, 18779, 1664, -4531, -302, 593, 28, -39, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 39*x^18 + 28*x^17 + 593*x^16 - 302*x^15 - 4531*x^14 + 1664*x^13 + 18779*x^12 - 5250*x^11 - 42734*x^10 + 10050*x^9 + 52063*x^8 - 11604*x^7 - 32303*x^6 + 7629*x^5 + 8990*x^4 - 2522*x^3 - 785*x^2 + 296*x - 23)
 
gp: K = bnfinit(x^20 - x^19 - 39*x^18 + 28*x^17 + 593*x^16 - 302*x^15 - 4531*x^14 + 1664*x^13 + 18779*x^12 - 5250*x^11 - 42734*x^10 + 10050*x^9 + 52063*x^8 - 11604*x^7 - 32303*x^6 + 7629*x^5 + 8990*x^4 - 2522*x^3 - 785*x^2 + 296*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 39 x^{18} + 28 x^{17} + 593 x^{16} - 302 x^{15} - 4531 x^{14} + 1664 x^{13} + 18779 x^{12} - 5250 x^{11} - 42734 x^{10} + 10050 x^{9} + 52063 x^{8} - 11604 x^{7} - 32303 x^{6} + 7629 x^{5} + 8990 x^{4} - 2522 x^{3} - 785 x^{2} + 296 x - 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7588612227812438544203562395366777=11^{16}\cdot 2777^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.43$
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:  $11, 2777$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{117249061491308935955701099637} a^{19} + \frac{255882062862717482617874355}{117249061491308935955701099637} a^{18} - \frac{3434946136040614678927557880}{117249061491308935955701099637} a^{17} - \frac{49691471835643198864512899497}{117249061491308935955701099637} a^{16} - \frac{53071504468326653302008636803}{117249061491308935955701099637} a^{15} - \frac{2664301972997687115533798582}{117249061491308935955701099637} a^{14} + \frac{44956308083362576703594188596}{117249061491308935955701099637} a^{13} + \frac{3302701846359431930538878787}{117249061491308935955701099637} a^{12} + \frac{50704801313388441015985391182}{117249061491308935955701099637} a^{11} + \frac{31089169099982194949795589764}{117249061491308935955701099637} a^{10} + \frac{25603027047145877956584183353}{117249061491308935955701099637} a^{9} - \frac{25669887413955530042944107967}{117249061491308935955701099637} a^{8} + \frac{26339323636537330651608988131}{117249061491308935955701099637} a^{7} - \frac{1210375444648707986635456402}{117249061491308935955701099637} a^{6} + \frac{11945319539490389097759902689}{117249061491308935955701099637} a^{5} + \frac{24941502965778983027106071397}{117249061491308935955701099637} a^{4} - \frac{28659921272594920696810178116}{117249061491308935955701099637} a^{3} + \frac{18778589878018902180005071510}{117249061491308935955701099637} a^{2} - \frac{46532136161063621345497651867}{117249061491308935955701099637} a + \frac{56149475804699671459630410955}{117249061491308935955701099637}$

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:  $19$
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:  \( 40125051391.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_4$ (as 20T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 25 conjugacy class representatives for $C_5\times S_4$
Character table for $C_5\times S_4$ is not computed

Intermediate fields

4.4.2777.1, \(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 $15{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $20$ $20$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
11Data not computed
2777Data not computed