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Properties

Label 20.20.1708837639...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{24}\cdot 5^{10}\cdot 7^{16}\cdot 11^{12}$
Root discriminant $102.72$
Ramified primes $2, 5, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T145

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -344, 2736, 51424, 115592, -212816, -578692, 358416, 977807, -284990, -738416, 90408, 258918, 574, -40692, -2762, 2735, 204, -84, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 84*x^18 + 204*x^17 + 2735*x^16 - 2762*x^15 - 40692*x^14 + 574*x^13 + 258918*x^12 + 90408*x^11 - 738416*x^10 - 284990*x^9 + 977807*x^8 + 358416*x^7 - 578692*x^6 - 212816*x^5 + 115592*x^4 + 51424*x^3 + 2736*x^2 - 344*x - 4)
 
gp: K = bnfinit(x^20 - 4*x^19 - 84*x^18 + 204*x^17 + 2735*x^16 - 2762*x^15 - 40692*x^14 + 574*x^13 + 258918*x^12 + 90408*x^11 - 738416*x^10 - 284990*x^9 + 977807*x^8 + 358416*x^7 - 578692*x^6 - 212816*x^5 + 115592*x^4 + 51424*x^3 + 2736*x^2 - 344*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 84 x^{18} + 204 x^{17} + 2735 x^{16} - 2762 x^{15} - 40692 x^{14} + 574 x^{13} + 258918 x^{12} + 90408 x^{11} - 738416 x^{10} - 284990 x^{9} + 977807 x^{8} + 358416 x^{7} - 578692 x^{6} - 212816 x^{5} + 115592 x^{4} + 51424 x^{3} + 2736 x^{2} - 344 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17088376396387871021245779312640000000000=2^{24}\cdot 5^{10}\cdot 7^{16}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.72$
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:  $2, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{18} + \frac{1}{20} a^{17} + \frac{1}{10} a^{16} + \frac{9}{40} a^{14} - \frac{1}{5} a^{13} + \frac{1}{10} a^{12} - \frac{1}{4} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{7}{20} a^{7} + \frac{19}{40} a^{6} + \frac{3}{20} a^{5} + \frac{1}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{20} a^{2} + \frac{1}{10}$, $\frac{1}{116627760686633834017396338296259574969407800} a^{19} + \frac{658551196497865153420033509739007261258607}{58313880343316917008698169148129787484703900} a^{18} + \frac{1267596749259224370046718772197352679438471}{14578470085829229252174542287032446871175975} a^{17} + \frac{2628032767915717476612100462912765066143957}{29156940171658458504349084574064893742351950} a^{16} + \frac{7121055673939492617817272504355663128803389}{116627760686633834017396338296259574969407800} a^{15} - \frac{1045290425613871086337991663698131485185963}{5831388034331691700869816914812978748470390} a^{14} - \frac{7258271946723395813908677208133229148220443}{29156940171658458504349084574064893742351950} a^{13} - \frac{6395990689204443616249507585082277908085311}{58313880343316917008698169148129787484703900} a^{12} - \frac{187721891424886801232717202580001600076891}{14578470085829229252174542287032446871175975} a^{11} - \frac{2934994884403612921206665143561021287161462}{14578470085829229252174542287032446871175975} a^{10} - \frac{10407300957703420755998757618873788175259111}{29156940171658458504349084574064893742351950} a^{9} - \frac{2837191282361608877377562403428943423565091}{58313880343316917008698169148129787484703900} a^{8} + \frac{22198517441605779027097170057546822916598931}{116627760686633834017396338296259574969407800} a^{7} + \frac{7761462553392243024538905719724714401827637}{58313880343316917008698169148129787484703900} a^{6} - \frac{2872072287266558096092637425161480829563203}{5831388034331691700869816914812978748470390} a^{5} + \frac{965247135925908692057172883795727835021138}{14578470085829229252174542287032446871175975} a^{4} + \frac{22855691978273752915219330216837365789472507}{58313880343316917008698169148129787484703900} a^{3} + \frac{2730917386201616103368659100147368057963322}{14578470085829229252174542287032446871175975} a^{2} + \frac{2028552440157925623870620454107825691579951}{29156940171658458504349084574064893742351950} a - \frac{5350969251809268249946994703013998585808684}{14578470085829229252174542287032446871175975}$

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:  $19$
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:  \( 82650511663700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T145:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for t20n145
Character table for t20n145

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.5228900671672832000.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 15 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
$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}$
$7$7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$