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Properties

Label 20.0.65425329716...9969.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{28}\cdot 7^{6}\cdot 79^{6}$
Root discriminant $30.96$
Ramified primes $3, 7, 79$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21, -147, 336, -84, -461, -2239, 14757, -40386, 72432, -96015, 99350, -82466, 55743, -30920, 14090, -5268, 1602, -390, 75, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21)
 
gp: K = bnfinit(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 75 x^{18} - 390 x^{17} + 1602 x^{16} - 5268 x^{15} + 14090 x^{14} - 30920 x^{13} + 55743 x^{12} - 82466 x^{11} + 99350 x^{10} - 96015 x^{9} + 72432 x^{8} - 40386 x^{7} + 14757 x^{6} - 2239 x^{5} - 461 x^{4} - 84 x^{3} + 336 x^{2} - 147 x + 21 \)

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:  \(654253297161106870378547679969=3^{28}\cdot 7^{6}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.96$
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:  $3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{4424737629} a^{18} - \frac{3}{1474912543} a^{17} - \frac{345077830}{4424737629} a^{16} - \frac{63067414}{1474912543} a^{15} + \frac{177722150}{1474912543} a^{14} - \frac{140375543}{1474912543} a^{13} - \frac{840658613}{4424737629} a^{12} + \frac{636632832}{1474912543} a^{11} + \frac{1365593618}{4424737629} a^{10} + \frac{237042511}{1474912543} a^{9} + \frac{246164580}{1474912543} a^{8} - \frac{246739417}{1474912543} a^{7} + \frac{284095381}{4424737629} a^{6} - \frac{641958939}{1474912543} a^{5} + \frac{1735667189}{4424737629} a^{4} - \frac{340344575}{1474912543} a^{3} - \frac{696368617}{1474912543} a^{2} + \frac{98085853}{1474912543} a + \frac{676125672}{1474912543}$, $\frac{1}{4424737629} a^{19} - \frac{345077911}{4424737629} a^{17} - \frac{345077626}{4424737629} a^{16} + \frac{305258815}{4424737629} a^{15} - \frac{15788736}{1474912543} a^{14} - \frac{560324396}{1474912543} a^{13} + \frac{243621151}{4424737629} a^{12} + \frac{855729566}{4424737629} a^{11} + \frac{1202169751}{4424737629} a^{10} + \frac{1238991365}{4424737629} a^{9} + \frac{493829260}{1474912543} a^{8} - \frac{478218706}{4424737629} a^{7} - \frac{843930931}{4424737629} a^{6} + \frac{2101726352}{4424737629} a^{5} + \frac{441919363}{1474912543} a^{4} + \frac{51174449}{113454811} a^{3} - \frac{269581528}{1474912543} a^{2} + \frac{83985806}{1474912543} a + \frac{185480876}{1474912543}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{35095174}{414573} a^{19} - \frac{333404153}{414573} a^{18} + \frac{821808535}{138191} a^{17} - \frac{4151437247}{138191} a^{16} + \frac{49994646140}{414573} a^{15} - \frac{53293611612}{138191} a^{14} + \frac{414538330490}{414573} a^{13} - \frac{292612254389}{138191} a^{12} + \frac{505767948756}{138191} a^{11} - \frac{2135338474280}{414573} a^{10} + \frac{2418782525449}{414573} a^{9} - \frac{719991559532}{138191} a^{8} + \frac{487255237592}{138191} a^{7} - \frac{686313903214}{414573} a^{6} + \frac{58230323547}{138191} a^{5} + \frac{8765386439}{414573} a^{4} - \frac{3930071607}{138191} a^{3} - \frac{2946519135}{138191} a^{2} + \frac{2456270922}{138191} a - \frac{491365827}{138191} \) (order $6$)
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:  \( 4740283.57365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.403137.1, 10.0.487558322307.1, 10.8.269619752235771.1, 10.2.808859256707313.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.22.44$x^{12} + 36 x^{11} - 27 x^{10} - 33 x^{9} - 18 x^{8} + 9 x^{7} - 24 x^{6} - 36 x^{3} - 27 x^{2} - 27 x + 36$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
$7$7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.3.2$x^{4} - 79$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.3.2$x^{4} - 79$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$