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Properties

Label 20.4.99718993574...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{8}\cdot 5^{10}\cdot 52501^{5}$
Root discriminant $44.66$
Ramified primes $2, 5, 52501$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![82031, -232758, 298031, -163979, -28636, 420297, -404468, 147378, 12898, -59331, 29423, -9470, 752, 775, -835, 336, -126, 0, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 126*x^16 + 336*x^15 - 835*x^14 + 775*x^13 + 752*x^12 - 9470*x^11 + 29423*x^10 - 59331*x^9 + 12898*x^8 + 147378*x^7 - 404468*x^6 + 420297*x^5 - 28636*x^4 - 163979*x^3 + 298031*x^2 - 232758*x + 82031)
 
gp: K = bnfinit(x^20 - 4*x^19 + 9*x^18 - 126*x^16 + 336*x^15 - 835*x^14 + 775*x^13 + 752*x^12 - 9470*x^11 + 29423*x^10 - 59331*x^9 + 12898*x^8 + 147378*x^7 - 404468*x^6 + 420297*x^5 - 28636*x^4 - 163979*x^3 + 298031*x^2 - 232758*x + 82031, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 9 x^{18} - 126 x^{16} + 336 x^{15} - 835 x^{14} + 775 x^{13} + 752 x^{12} - 9470 x^{11} + 29423 x^{10} - 59331 x^{9} + 12898 x^{8} + 147378 x^{7} - 404468 x^{6} + 420297 x^{5} - 28636 x^{4} - 163979 x^{3} + 298031 x^{2} - 232758 x + 82031 \)

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:  \(997189935746553281906252500000000=2^{8}\cdot 5^{10}\cdot 52501^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.66$
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:  $2, 5, 52501$
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}{221301397043161367188249691233614393967009142322305070723442} a^{19} + \frac{82952729357248774766465549432892635353864644835014444468947}{221301397043161367188249691233614393967009142322305070723442} a^{18} - \frac{5210899563618158641961003535937489030728731534438671888787}{110650698521580683594124845616807196983504571161152535361721} a^{17} + \frac{24852162891482022441685924654743141375073393940568261422991}{110650698521580683594124845616807196983504571161152535361721} a^{16} - \frac{9128958622342365221568309469022959911610710405294461072200}{110650698521580683594124845616807196983504571161152535361721} a^{15} + \frac{34126029762482356681021236008189234029444590214063503773956}{110650698521580683594124845616807196983504571161152535361721} a^{14} - \frac{469681060001014982126491233940203137613455856632093449723}{2068237355543564179329436366669293401560833105815935240406} a^{13} + \frac{32954111871292598411917073663690223899341610893493832621176}{110650698521580683594124845616807196983504571161152535361721} a^{12} + \frac{36448180785993714248634978095234569528057292445773569697465}{110650698521580683594124845616807196983504571161152535361721} a^{11} + \frac{53055263448270924115219114839779006527535946498662272265467}{110650698521580683594124845616807196983504571161152535361721} a^{10} + \frac{24489543383621115080180361046506214355921705086308086791391}{221301397043161367188249691233614393967009142322305070723442} a^{9} - \frac{10690474441413565902547222227169580049494196842475537213582}{110650698521580683594124845616807196983504571161152535361721} a^{8} + \frac{1142580619024127124852566085852862363643897807756900428249}{10059154411052789417647713237891563362136779196468412305611} a^{7} - \frac{4126055431255466991817693178378841082543843250200293894953}{10059154411052789417647713237891563362136779196468412305611} a^{6} - \frac{3185083225961564493750975263602517494777598485936173936898}{110650698521580683594124845616807196983504571161152535361721} a^{5} + \frac{33195694288159799888680608795321887362713853359987464947539}{221301397043161367188249691233614393967009142322305070723442} a^{4} + \frac{46482720064372720575465671144042640323926165251992993842747}{221301397043161367188249691233614393967009142322305070723442} a^{3} - \frac{18723511173638237999888095573356395085972985548188389021555}{110650698521580683594124845616807196983504571161152535361721} a^{2} + \frac{10276217459296535046648925472869691715262040396231820129021}{221301397043161367188249691233614393967009142322305070723442} a - \frac{22884353411927718789155930013460117205059393645768066611693}{221301397043161367188249691233614393967009142322305070723442}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 249759139.276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8613609378125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
52501Data not computed