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Properties

Label 20.4.92975432835...9136.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 14057^{2}$
Root discriminant $25.03$
Ramified primes $2, 11, 14057$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, -11, -264, -738, -945, -229, 343, 92, -395, 81, 136, -137, -123, 115, -14, -38, 29, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 4*x^18 + 29*x^17 - 38*x^16 - 14*x^15 + 115*x^14 - 123*x^13 - 137*x^12 + 136*x^11 + 81*x^10 - 395*x^9 + 92*x^8 + 343*x^7 - 229*x^6 - 945*x^5 - 738*x^4 - 264*x^3 - 11*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 4*x^18 + 29*x^17 - 38*x^16 - 14*x^15 + 115*x^14 - 123*x^13 - 137*x^12 + 136*x^11 + 81*x^10 - 395*x^9 + 92*x^8 + 343*x^7 - 229*x^6 - 945*x^5 - 738*x^4 - 264*x^3 - 11*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 4 x^{18} + 29 x^{17} - 38 x^{16} - 14 x^{15} + 115 x^{14} - 123 x^{13} - 137 x^{12} + 136 x^{11} + 81 x^{10} - 395 x^{9} + 92 x^{8} + 343 x^{7} - 229 x^{6} - 945 x^{5} - 738 x^{4} - 264 x^{3} - 11 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9297543283501805026208859136=2^{10}\cdot 11^{16}\cdot 14057^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.03$
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:  $2, 11, 14057$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2523067641821833127194032773} a^{19} - \frac{58657761619480598408606060}{2523067641821833127194032773} a^{18} + \frac{995493605171321515436803975}{2523067641821833127194032773} a^{17} + \frac{547044253101371483845004212}{2523067641821833127194032773} a^{16} + \frac{867032561460684645074170963}{2523067641821833127194032773} a^{15} + \frac{629620674699148140660478592}{2523067641821833127194032773} a^{14} - \frac{486482565120671056883760761}{2523067641821833127194032773} a^{13} - \frac{240371997144071762028370495}{2523067641821833127194032773} a^{12} + \frac{1185751379047806970952283509}{2523067641821833127194032773} a^{11} + \frac{417932458993424838877256081}{2523067641821833127194032773} a^{10} - \frac{1190347924831978725360840021}{2523067641821833127194032773} a^{9} - \frac{979297536515181193516112011}{2523067641821833127194032773} a^{8} + \frac{232013674706140316701666769}{2523067641821833127194032773} a^{7} - \frac{45174413547269231822184323}{2523067641821833127194032773} a^{6} + \frac{1048867415223394430211404131}{2523067641821833127194032773} a^{5} - \frac{260095370597378703500006396}{2523067641821833127194032773} a^{4} + \frac{473481130682298478112838634}{2523067641821833127194032773} a^{3} - \frac{448739463391666637818203762}{2523067641821833127194032773} a^{2} - \frac{365128664398008570357752124}{2523067641821833127194032773} a + \frac{61861544488717077819066693}{2523067641821833127194032773}$

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:  $11$
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:  \( 990804.459375 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.3013242790217.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
14057Data not computed