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Properties

Label 20.0.17406652163...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{11}\cdot 19^{6}\cdot 461^{4}$
Root discriminant $45.92$
Ramified primes $2, 5, 19, 461$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5345, -22760, 61531, -24494, 41709, -34124, 31556, -79844, 85675, -68258, 48551, -11284, -3341, -848, 1192, 12, 19, -40, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 - 40*x^17 + 19*x^16 + 12*x^15 + 1192*x^14 - 848*x^13 - 3341*x^12 - 11284*x^11 + 48551*x^10 - 68258*x^9 + 85675*x^8 - 79844*x^7 + 31556*x^6 - 34124*x^5 + 41709*x^4 - 24494*x^3 + 61531*x^2 - 22760*x + 5345)
 
gp: K = bnfinit(x^20 - 3*x^18 - 40*x^17 + 19*x^16 + 12*x^15 + 1192*x^14 - 848*x^13 - 3341*x^12 - 11284*x^11 + 48551*x^10 - 68258*x^9 + 85675*x^8 - 79844*x^7 + 31556*x^6 - 34124*x^5 + 41709*x^4 - 24494*x^3 + 61531*x^2 - 22760*x + 5345, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} - 40 x^{17} + 19 x^{16} + 12 x^{15} + 1192 x^{14} - 848 x^{13} - 3341 x^{12} - 11284 x^{11} + 48551 x^{10} - 68258 x^{9} + 85675 x^{8} - 79844 x^{7} + 31556 x^{6} - 34124 x^{5} + 41709 x^{4} - 24494 x^{3} + 61531 x^{2} - 22760 x + 5345 \)

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:  \(1740665216320641860403200000000000=2^{24}\cdot 5^{11}\cdot 19^{6}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.92$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{15} a^{18} - \frac{7}{15} a^{17} + \frac{1}{3} a^{16} + \frac{2}{15} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{7}{15} a^{12} - \frac{2}{15} a^{11} + \frac{1}{15} a^{10} - \frac{4}{15} a^{9} - \frac{2}{15} a^{8} + \frac{1}{3} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{3} a^{4} - \frac{1}{15} a^{3} + \frac{2}{5} a^{2} - \frac{1}{3}$, $\frac{1}{155596933368010605461361335234321613331272945501965475} a^{19} - \frac{733988277168737464063503361646270424954702323352247}{51865644456003535153787111744773871110424315167321825} a^{18} - \frac{35358077276249833603658170212325871495867408797105247}{155596933368010605461361335234321613331272945501965475} a^{17} + \frac{45164997206071411165054247152251110623829607814998487}{155596933368010605461361335234321613331272945501965475} a^{16} - \frac{2908270544264614254982683636677388827356619212174941}{51865644456003535153787111744773871110424315167321825} a^{15} + \frac{11426096129564655200267468581492010620329822351968881}{31119386673602121092272267046864322666254589100393095} a^{14} - \frac{77594706508641796518370760909582812829149265645224888}{155596933368010605461361335234321613331272945501965475} a^{13} - \frac{7205022767823129482843958254165694479742930348454073}{31119386673602121092272267046864322666254589100393095} a^{12} - \frac{34828950609671385956312779489061355565510094498553901}{155596933368010605461361335234321613331272945501965475} a^{11} - \frac{4457160181913908485985240602937433914188386050757606}{51865644456003535153787111744773871110424315167321825} a^{10} + \frac{14143931042426781286026150778246863885815410898205963}{51865644456003535153787111744773871110424315167321825} a^{9} - \frac{9707346116872788115186292752653132835092693484673319}{51865644456003535153787111744773871110424315167321825} a^{8} - \frac{47957625091709817723356923960775528476948694555442313}{155596933368010605461361335234321613331272945501965475} a^{7} - \frac{21097877065160998493774065654103014716451113845825987}{51865644456003535153787111744773871110424315167321825} a^{6} + \frac{1159709960698464984012082610560124086682941140212282}{155596933368010605461361335234321613331272945501965475} a^{5} - \frac{48104904501722417301920789821389911410071755526355511}{155596933368010605461361335234321613331272945501965475} a^{4} + \frac{2192208206933389145332005563383808432696742137976562}{31119386673602121092272267046864322666254589100393095} a^{3} - \frac{16545745037519513436657206627825692894965607816773443}{51865644456003535153787111744773871110424315167321825} a^{2} + \frac{14604250247908745453425956493273530131514649237363574}{31119386673602121092272267046864322666254589100393095} a - \frac{4019334175846108797154752123058197840090459873063706}{31119386673602121092272267046864322666254589100393095}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -1 \) (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:  \( 134075984.499 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed