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Properties

Label 20.4.31782027227...0849.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{12}\cdot 97^{2}\cdot 347^{4}$
Root discriminant $23.72$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 17, -279, 1968, -4996, 8066, -8788, 7488, -5272, 3797, -2715, 1764, -865, 390, -207, 84, -31, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 7*x^18 - 31*x^17 + 84*x^16 - 207*x^15 + 390*x^14 - 865*x^13 + 1764*x^12 - 2715*x^11 + 3797*x^10 - 5272*x^9 + 7488*x^8 - 8788*x^7 + 8066*x^6 - 4996*x^5 + 1968*x^4 - 279*x^3 + 17*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 7*x^18 - 31*x^17 + 84*x^16 - 207*x^15 + 390*x^14 - 865*x^13 + 1764*x^12 - 2715*x^11 + 3797*x^10 - 5272*x^9 + 7488*x^8 - 8788*x^7 + 8066*x^6 - 4996*x^5 + 1968*x^4 - 279*x^3 + 17*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 7 x^{18} - 31 x^{17} + 84 x^{16} - 207 x^{15} + 390 x^{14} - 865 x^{13} + 1764 x^{12} - 2715 x^{11} + 3797 x^{10} - 5272 x^{9} + 7488 x^{8} - 8788 x^{7} + 8066 x^{6} - 4996 x^{5} + 1968 x^{4} - 279 x^{3} + 17 x^{2} - 8 x + 1 \)

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:  \(3178202722755606120397450849=13^{12}\cdot 97^{2}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.72$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{101} a^{18} + \frac{48}{101} a^{17} + \frac{29}{101} a^{16} + \frac{39}{101} a^{15} - \frac{5}{101} a^{14} - \frac{35}{101} a^{13} + \frac{29}{101} a^{12} - \frac{23}{101} a^{11} + \frac{28}{101} a^{10} - \frac{29}{101} a^{9} + \frac{40}{101} a^{8} - \frac{43}{101} a^{7} - \frac{37}{101} a^{6} + \frac{16}{101} a^{5} - \frac{33}{101} a^{4} - \frac{45}{101} a^{3} + \frac{42}{101} a^{2} + \frac{34}{101} a - \frac{50}{101}$, $\frac{1}{547392867751467610714531337} a^{19} - \frac{2258982940150215116455909}{547392867751467610714531337} a^{18} - \frac{219760472833387040405597325}{547392867751467610714531337} a^{17} + \frac{187803891307553730757503804}{547392867751467610714531337} a^{16} - \frac{29063353875791868274432143}{547392867751467610714531337} a^{15} + \frac{224028411156464958427272606}{547392867751467610714531337} a^{14} + \frac{224279215538758802723439941}{547392867751467610714531337} a^{13} + \frac{140589773452089909107249654}{547392867751467610714531337} a^{12} + \frac{234175000389069185601541461}{547392867751467610714531337} a^{11} + \frac{92328646595680396640057875}{547392867751467610714531337} a^{10} + \frac{207138223280600669158648302}{547392867751467610714531337} a^{9} + \frac{55151314478835580017932444}{547392867751467610714531337} a^{8} + \frac{233378015308589693877598437}{547392867751467610714531337} a^{7} + \frac{168414697631809269525759504}{547392867751467610714531337} a^{6} + \frac{4255594653937746787967591}{13351045554913844163769057} a^{5} + \frac{20850877395736762209901268}{547392867751467610714531337} a^{4} - \frac{135962040859048141050635242}{547392867751467610714531337} a^{3} + \frac{197529840232263715400672594}{547392867751467610714531337} a^{2} - \frac{433812329638072621210876}{547392867751467610714531337} a + \frac{244952399450500236119230864}{547392867751467610714531337}$

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:  \( 639667.330196 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.2.4336580827189.1, 10.2.333583140553.1, 10.6.44707018837.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed