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Properties

Label 20.8.18358429263...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $41.04$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7975, 33350, -33350, 14515, -40151, 49786, -21146, 18640, -18777, 6844, -6138, 4651, 1469, -2188, 434, -44, 35, 32, -18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 18*x^18 + 32*x^17 + 35*x^16 - 44*x^15 + 434*x^14 - 2188*x^13 + 1469*x^12 + 4651*x^11 - 6138*x^10 + 6844*x^9 - 18777*x^8 + 18640*x^7 - 21146*x^6 + 49786*x^5 - 40151*x^4 + 14515*x^3 - 33350*x^2 + 33350*x - 7975)
 
gp: K = bnfinit(x^20 - x^19 - 18*x^18 + 32*x^17 + 35*x^16 - 44*x^15 + 434*x^14 - 2188*x^13 + 1469*x^12 + 4651*x^11 - 6138*x^10 + 6844*x^9 - 18777*x^8 + 18640*x^7 - 21146*x^6 + 49786*x^5 - 40151*x^4 + 14515*x^3 - 33350*x^2 + 33350*x - 7975, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 18 x^{18} + 32 x^{17} + 35 x^{16} - 44 x^{15} + 434 x^{14} - 2188 x^{13} + 1469 x^{12} + 4651 x^{11} - 6138 x^{10} + 6844 x^{9} - 18777 x^{8} + 18640 x^{7} - 21146 x^{6} + 49786 x^{5} - 40151 x^{4} + 14515 x^{3} - 33350 x^{2} + 33350 x - 7975 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(183584292633496284375040000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.04$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{12965701287666418922884495121440087992111965} a^{19} + \frac{1092737368755986849689884651833547438456712}{12965701287666418922884495121440087992111965} a^{18} + \frac{1562814034664603919968465797799152085194529}{12965701287666418922884495121440087992111965} a^{17} - \frac{3173560417972954978940069441472980369975722}{12965701287666418922884495121440087992111965} a^{16} - \frac{1988749974513556981007870467671306939272749}{12965701287666418922884495121440087992111965} a^{15} - \frac{1260854595416904409746205165444615927119534}{12965701287666418922884495121440087992111965} a^{14} + \frac{1292522463889283436824504336889029966137987}{12965701287666418922884495121440087992111965} a^{13} - \frac{4209406835871860682907048377875106400662786}{12965701287666418922884495121440087992111965} a^{12} + \frac{853695471880669421428952134229125618953667}{2593140257533283784576899024288017598422393} a^{11} - \frac{2244751301519317216704668145680013727703557}{12965701287666418922884495121440087992111965} a^{10} - \frac{669662178723233522483322148386432772866926}{2593140257533283784576899024288017598422393} a^{9} - \frac{975107710255643885648632508488673738635559}{2593140257533283784576899024288017598422393} a^{8} - \frac{109183074299442028548373747820741688919379}{2593140257533283784576899024288017598422393} a^{7} - \frac{6306809739183452847463423555556375051600076}{12965701287666418922884495121440087992111965} a^{6} + \frac{1802504353838249029847779516349504767955789}{12965701287666418922884495121440087992111965} a^{5} + \frac{2061749478652155142340682771825957041981463}{12965701287666418922884495121440087992111965} a^{4} + \frac{59125294819330356329423763457864382269426}{350424359126119430348229597876759134921945} a^{3} + \frac{4008254185504615273937049330575066632189607}{12965701287666418922884495121440087992111965} a^{2} + \frac{710035449279574814089478937784451923198546}{2593140257533283784576899024288017598422393} a + \frac{1223580347660466318441298841113338036943112}{2593140257533283784576899024288017598422393}$

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:  $13$
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:  \( 822626143.441 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
31Data not computed