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Properties

Label 20.0.62768567534...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 11^{12}$
Root discriminant $24.54$
Ramified primes $2, 5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 154, 572, -693, -2189, 2062, 2070, -2984, 1242, 442, -454, -2148, 2957, -856, -284, 21, 84, 8, -14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 14*x^18 + 8*x^17 + 84*x^16 + 21*x^15 - 284*x^14 - 856*x^13 + 2957*x^12 - 2148*x^11 - 454*x^10 + 442*x^9 + 1242*x^8 - 2984*x^7 + 2070*x^6 + 2062*x^5 - 2189*x^4 - 693*x^3 + 572*x^2 + 154*x + 11)
 
gp: K = bnfinit(x^20 - x^19 - 14*x^18 + 8*x^17 + 84*x^16 + 21*x^15 - 284*x^14 - 856*x^13 + 2957*x^12 - 2148*x^11 - 454*x^10 + 442*x^9 + 1242*x^8 - 2984*x^7 + 2070*x^6 + 2062*x^5 - 2189*x^4 - 693*x^3 + 572*x^2 + 154*x + 11, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 14 x^{18} + 8 x^{17} + 84 x^{16} + 21 x^{15} - 284 x^{14} - 856 x^{13} + 2957 x^{12} - 2148 x^{11} - 454 x^{10} + 442 x^{9} + 1242 x^{8} - 2984 x^{7} + 2070 x^{6} + 2062 x^{5} - 2189 x^{4} - 693 x^{3} + 572 x^{2} + 154 x + 11 \)

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:  \(6276856753442000000000000000=2^{16}\cdot 5^{15}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.54$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{166894398778707808064542466939} a^{19} - \frac{81342462414878754472680703001}{166894398778707808064542466939} a^{18} + \frac{82975572822813404378177336527}{166894398778707808064542466939} a^{17} - \frac{32613089877683084937381871053}{166894398778707808064542466939} a^{16} + \frac{34645633792121937635007252191}{166894398778707808064542466939} a^{15} + \frac{71329178422629680643197193156}{166894398778707808064542466939} a^{14} + \frac{48122217081419285585794148776}{166894398778707808064542466939} a^{13} + \frac{50321614987955554498680341230}{166894398778707808064542466939} a^{12} - \frac{42899926787157619501645068786}{166894398778707808064542466939} a^{11} + \frac{76379818171470440054247500927}{166894398778707808064542466939} a^{10} + \frac{50748945222538462762456310419}{166894398778707808064542466939} a^{9} + \frac{67177527661623251960125733291}{166894398778707808064542466939} a^{8} - \frac{19507557483348244984263697412}{166894398778707808064542466939} a^{7} + \frac{60516221341276761211800597169}{166894398778707808064542466939} a^{6} - \frac{29679206436944698872352897170}{166894398778707808064542466939} a^{5} + \frac{81849202144785070030539845747}{166894398778707808064542466939} a^{4} - \frac{24547693278905361901431792595}{166894398778707808064542466939} a^{3} - \frac{50669377913887003485321598599}{166894398778707808064542466939} a^{2} - \frac{28662932791678913525008537462}{166894398778707808064542466939} a - \frac{13128490378503943648289869599}{166894398778707808064542466939}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{7289060764984650835005400}{43678199104608167512311559} a^{19} + \frac{353858640310315206658040}{43678199104608167512311559} a^{18} + \frac{100402197392328906938671128}{43678199104608167512311559} a^{17} + \frac{36831152220859970975538457}{43678199104608167512311559} a^{16} - \frac{550348032519642192977979384}{43678199104608167512311559} a^{15} - \frac{660494380548773547570386076}{43678199104608167512311559} a^{14} + \frac{1299722710662715253733749416}{43678199104608167512311559} a^{13} + \frac{7267615178063123569779326144}{43678199104608167512311559} a^{12} - \frac{14351607598008853506725261940}{43678199104608167512311559} a^{11} + \frac{4004709282481430988047733996}{43678199104608167512311559} a^{10} + \frac{3705311903005180561338399376}{43678199104608167512311559} a^{9} + \frac{855135795348139793562569126}{43678199104608167512311559} a^{8} - \frac{7383438263957563805663293836}{43678199104608167512311559} a^{7} + \frac{15148777819431723639328561640}{43678199104608167512311559} a^{6} - \frac{2616110873826516385385280640}{43678199104608167512311559} a^{5} - \frac{13791710927598537393170991940}{43678199104608167512311559} a^{4} + \frac{2772122559360190350359044692}{43678199104608167512311559} a^{3} + \frac{4315266349251662694131239052}{43678199104608167512311559} a^{2} + \frac{4101051333996469108770932}{43678199104608167512311559} a - \frac{63937500830760514518040451}{43678199104608167512311559} \) (order $10$)
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:  \( 475214.0571061137 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 25 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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$