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Properties

Label 20.0.72694810795...4441.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{26}\cdot 7^{6}\cdot 79^{6}$
Root discriminant $27.74$
Ramified primes $3, 7, 79$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T288

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, -656, 2272, -4565, 6930, -8533, 6404, -2700, 3830, -3962, 324, 338, 731, -105, -373, 70, 54, -7, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + x^18 - 7*x^17 + 54*x^16 + 70*x^15 - 373*x^14 - 105*x^13 + 731*x^12 + 338*x^11 + 324*x^10 - 3962*x^9 + 3830*x^8 - 2700*x^7 + 6404*x^6 - 8533*x^5 + 6930*x^4 - 4565*x^3 + 2272*x^2 - 656*x + 79)
 
gp: K = bnfinit(x^20 - 4*x^19 + x^18 - 7*x^17 + 54*x^16 + 70*x^15 - 373*x^14 - 105*x^13 + 731*x^12 + 338*x^11 + 324*x^10 - 3962*x^9 + 3830*x^8 - 2700*x^7 + 6404*x^6 - 8533*x^5 + 6930*x^4 - 4565*x^3 + 2272*x^2 - 656*x + 79, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + x^{18} - 7 x^{17} + 54 x^{16} + 70 x^{15} - 373 x^{14} - 105 x^{13} + 731 x^{12} + 338 x^{11} + 324 x^{10} - 3962 x^{9} + 3830 x^{8} - 2700 x^{7} + 6404 x^{6} - 8533 x^{5} + 6930 x^{4} - 4565 x^{3} + 2272 x^{2} - 656 x + 79 \)

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:  \(72694810795678541153171964441=3^{26}\cdot 7^{6}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.74$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{15} a^{18} + \frac{4}{15} a^{17} + \frac{4}{15} a^{16} - \frac{1}{15} a^{15} + \frac{1}{3} a^{14} + \frac{4}{15} a^{13} - \frac{2}{5} a^{12} + \frac{1}{15} a^{11} - \frac{2}{15} a^{10} - \frac{7}{15} a^{9} - \frac{4}{15} a^{8} + \frac{4}{15} a^{7} + \frac{1}{5} a^{6} - \frac{2}{15} a^{5} + \frac{1}{15} a^{4} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2} - \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{1223539405370554230812714000145} a^{19} - \frac{4176515271371187459731083310}{244707881074110846162542800029} a^{18} + \frac{381559030992972991118601716498}{1223539405370554230812714000145} a^{17} - \frac{120155439080202710809630795024}{407846468456851410270904666715} a^{16} - \frac{534136302777387181858821469436}{1223539405370554230812714000145} a^{15} + \frac{174131902940882191977373040638}{407846468456851410270904666715} a^{14} - \frac{19167699421646267313078916952}{1223539405370554230812714000145} a^{13} - \frac{110653630713636525646658964382}{244707881074110846162542800029} a^{12} + \frac{506797673034322695807514601069}{1223539405370554230812714000145} a^{11} - \frac{103737191131555923418225220553}{407846468456851410270904666715} a^{10} - \frac{562129729494268594054399056131}{1223539405370554230812714000145} a^{9} - \frac{15678791449485368611055924198}{81569293691370282054180933343} a^{8} + \frac{13125047272682144496017245162}{39469013076469491316539161295} a^{7} - \frac{451330456403745075826216379849}{1223539405370554230812714000145} a^{6} - \frac{158186450140941029245710066571}{1223539405370554230812714000145} a^{5} - \frac{90741554380218309354654348557}{407846468456851410270904666715} a^{4} + \frac{392297238414112514381413462618}{1223539405370554230812714000145} a^{3} - \frac{73691910907683261415728932884}{407846468456851410270904666715} a^{2} + \frac{193526972215093949545317048778}{407846468456851410270904666715} a - \frac{47679601822249632252428956474}{1223539405370554230812714000145}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{205998706160402654}{4916126721362553867} a^{19} + \frac{271944275716685410}{1638708907120851289} a^{18} - \frac{45817462078843564}{1638708907120851289} a^{17} + \frac{1320830345905467313}{4916126721362553867} a^{16} - \frac{3714930861287760260}{1638708907120851289} a^{15} - \frac{15136559885131310602}{4916126721362553867} a^{14} + \frac{78023948138981063537}{4916126721362553867} a^{13} + \frac{28880266930177117108}{4916126721362553867} a^{12} - \frac{53387960998961892023}{1638708907120851289} a^{11} - \frac{90531192239178416240}{4916126721362553867} a^{10} - \frac{17630013688420192155}{1638708907120851289} a^{9} + \frac{846949367996063854925}{4916126721362553867} a^{8} - \frac{23209746976338726280}{158584732947179157} a^{7} + \frac{402940532635225886719}{4916126721362553867} a^{6} - \frac{427955370237596391387}{1638708907120851289} a^{5} + \frac{1659480679044012687970}{4916126721362553867} a^{4} - \frac{390175294511135488447}{1638708907120851289} a^{3} + \frac{759790407244874731580}{4916126721362553867} a^{2} - \frac{349136853159948439292}{4916126721362553867} a + \frac{67894278088604743450}{4916126721362553867} \) (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:  \( 6605869.44451 \) (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, 10.8.269619752235771.2, 10.0.487558322307.1, 10.2.89873250745257.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{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$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$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}$
79.4.3.2$x^{4} - 79$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$