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Properties

Label 20.0.17447867698...2544.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{52}\cdot 3^{18}$
Root discriminant $16.30$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -76, 155, -72, -291, 792, -1056, 744, 222, -1528, 2590, -2960, 2622, -1896, 1152, -600, 273, -108, 35, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 35*x^18 - 108*x^17 + 273*x^16 - 600*x^15 + 1152*x^14 - 1896*x^13 + 2622*x^12 - 2960*x^11 + 2590*x^10 - 1528*x^9 + 222*x^8 + 744*x^7 - 1056*x^6 + 792*x^5 - 291*x^4 - 72*x^3 + 155*x^2 - 76*x + 13)
 
gp: K = bnfinit(x^20 - 8*x^19 + 35*x^18 - 108*x^17 + 273*x^16 - 600*x^15 + 1152*x^14 - 1896*x^13 + 2622*x^12 - 2960*x^11 + 2590*x^10 - 1528*x^9 + 222*x^8 + 744*x^7 - 1056*x^6 + 792*x^5 - 291*x^4 - 72*x^3 + 155*x^2 - 76*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 35 x^{18} - 108 x^{17} + 273 x^{16} - 600 x^{15} + 1152 x^{14} - 1896 x^{13} + 2622 x^{12} - 2960 x^{11} + 2590 x^{10} - 1528 x^{9} + 222 x^{8} + 744 x^{7} - 1056 x^{6} + 792 x^{5} - 291 x^{4} - 72 x^{3} + 155 x^{2} - 76 x + 13 \)

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:  \(1744786769896095344492544=2^{52}\cdot 3^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.30$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{2} a^{11} - \frac{3}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{3}{8} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{8} a^{7} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{3}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{22837884484472} a^{19} - \frac{648189599921}{22837884484472} a^{18} + \frac{23815092338}{2854735560559} a^{17} + \frac{325905244433}{5709471121118} a^{16} + \frac{60356691283}{22837884484472} a^{15} + \frac{1951511940713}{22837884484472} a^{14} - \frac{2253478504959}{22837884484472} a^{13} + \frac{1485935973347}{22837884484472} a^{12} - \frac{9411385222135}{22837884484472} a^{11} - \frac{7805692923837}{22837884484472} a^{10} + \frac{10847374247619}{22837884484472} a^{9} + \frac{1431479777949}{22837884484472} a^{8} - \frac{3641437786293}{22837884484472} a^{7} - \frac{10376380026791}{22837884484472} a^{6} - \frac{3371920470047}{22837884484472} a^{5} + \frac{7668153532787}{22837884484472} a^{4} + \frac{13324002044}{69627696599} a^{3} - \frac{60181682287}{5709471121118} a^{2} - \frac{2762776400629}{22837884484472} a + \frac{8748074523881}{22837884484472}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{56551783397}{156423866332} a^{19} + \frac{431518190789}{156423866332} a^{18} - \frac{913161177855}{78211933166} a^{17} + \frac{10942480810011}{312847732664} a^{16} - \frac{6776010207235}{78211933166} a^{15} + \frac{58529602371917}{312847732664} a^{14} - \frac{55101190611777}{156423866332} a^{13} + \frac{88371467168931}{156423866332} a^{12} - \frac{118129749396401}{156423866332} a^{11} + \frac{254175759515141}{312847732664} a^{10} - \frac{25746266558645}{39105966583} a^{9} + \frac{25380587783423}{78211933166} a^{8} + \frac{2879482551785}{78211933166} a^{7} - \frac{82742614775803}{312847732664} a^{6} + \frac{46667052637415}{156423866332} a^{5} - \frac{29528389529477}{156423866332} a^{4} + \frac{40064660773}{953804063} a^{3} + \frac{13132092417803}{312847732664} a^{2} - \frac{3401687001943}{78211933166} a + \frac{3958560732185}{312847732664} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 69213.1676374 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 5.1.165888.1, 10.2.1320903770112.1, 10.0.440301256704.1, 10.0.82556485632.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
3Data not computed