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Properties

Label 20.14.5823779570...0000.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{4}\cdot 5^{15}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}$
Root discriminant $61.41$
Ramified primes $2, 5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24256, 55104, -145144, -374268, 14175, 721433, 500144, -406700, -455873, 56970, 157102, 25240, -28505, -9501, 3451, 844, -247, 30, -2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 2*x^18 + 30*x^17 - 247*x^16 + 844*x^15 + 3451*x^14 - 9501*x^13 - 28505*x^12 + 25240*x^11 + 157102*x^10 + 56970*x^9 - 455873*x^8 - 406700*x^7 + 500144*x^6 + 721433*x^5 + 14175*x^4 - 374268*x^3 - 145144*x^2 + 55104*x + 24256)
 
gp: K = bnfinit(x^20 - 5*x^19 - 2*x^18 + 30*x^17 - 247*x^16 + 844*x^15 + 3451*x^14 - 9501*x^13 - 28505*x^12 + 25240*x^11 + 157102*x^10 + 56970*x^9 - 455873*x^8 - 406700*x^7 + 500144*x^6 + 721433*x^5 + 14175*x^4 - 374268*x^3 - 145144*x^2 + 55104*x + 24256, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 2 x^{18} + 30 x^{17} - 247 x^{16} + 844 x^{15} + 3451 x^{14} - 9501 x^{13} - 28505 x^{12} + 25240 x^{11} + 157102 x^{10} + 56970 x^{9} - 455873 x^{8} - 406700 x^{7} + 500144 x^{6} + 721433 x^{5} + 14175 x^{4} - 374268 x^{3} - 145144 x^{2} + 55104 x + 24256 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-582377957040530370412846191406250000=-\,2^{4}\cdot 5^{15}\cdot 19^{4}\cdot 29^{6}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.41$
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:  $2, 5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{18} + \frac{1}{8} a^{17} - \frac{5}{12} a^{16} - \frac{1}{12} a^{15} + \frac{3}{8} a^{14} - \frac{1}{6} a^{13} + \frac{11}{24} a^{12} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{3} a^{9} + \frac{1}{4} a^{8} + \frac{5}{12} a^{7} - \frac{3}{8} a^{6} - \frac{1}{6} a^{5} + \frac{1}{24} a^{3} - \frac{1}{24} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{780225573390513354236341622635399061472583782490512} a^{19} + \frac{4363368960434236503378482405636329247400942596573}{780225573390513354236341622635399061472583782490512} a^{18} + \frac{16889927963603937181816891103803146010962712303181}{195056393347628338559085405658849765368145945622628} a^{17} - \frac{32525372382329625094769511702033453367195154234003}{390112786695256677118170811317699530736291891245256} a^{16} - \frac{347272885564231020712820571427023769558832221891291}{780225573390513354236341622635399061472583782490512} a^{15} + \frac{2670783253088986356181392949496537872003509716983}{390112786695256677118170811317699530736291891245256} a^{14} + \frac{1349634933652986617709510343914195596981030802945}{260075191130171118078780540878466353824194594163504} a^{13} - \frac{276644530578815334672654059272739253672761756358455}{780225573390513354236341622635399061472583782490512} a^{12} + \frac{78897666934959836025238147960213523293136055371791}{260075191130171118078780540878466353824194594163504} a^{11} + \frac{164003507288343574606410281603587532450878686356315}{390112786695256677118170811317699530736291891245256} a^{10} + \frac{106874141702718879930775699285726881791478648388567}{390112786695256677118170811317699530736291891245256} a^{9} + \frac{66320258152440847691241714906753462602410687810763}{390112786695256677118170811317699530736291891245256} a^{8} + \frac{112124100961805932393905177873172364411573227723715}{780225573390513354236341622635399061472583782490512} a^{7} + \frac{46521187514995555763107047081702032077552631105785}{390112786695256677118170811317699530736291891245256} a^{6} - \frac{22704241304704246885067250219331767333683218394593}{97528196673814169279542702829424882684072972811314} a^{5} - \frac{293573328614698985724419461369796485443349741869623}{780225573390513354236341622635399061472583782490512} a^{4} + \frac{276952789565221564725862133188897913088618036303505}{780225573390513354236341622635399061472583782490512} a^{3} - \frac{63677612861264933327627748081878649700205345837741}{130037595565085559039390270439233176912097297081752} a^{2} - \frac{6212239119826075894965365830862066655826455376557}{16254699445635694879923783804904147114012162135219} a - \frac{2999074041860333435229134007478220827255542031115}{48764098336907084639771351414712441342036486405657}$

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:  $16$
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:  \( 113145783806 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R $20$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
109Data not computed