↑

Properties

Label 20.8.70638883973...9408.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-347, 962, 612, -2944, 2774, 5314, -4056, -4924, 4883, 2496, -7776, -1914, 2818, 2528, -1572, -356, 324, -56, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 - 56*x^17 + 324*x^16 - 356*x^15 - 1572*x^14 + 2528*x^13 + 2818*x^12 - 1914*x^11 - 7776*x^10 + 2496*x^9 + 4883*x^8 - 4924*x^7 - 4056*x^6 + 5314*x^5 + 2774*x^4 - 2944*x^3 + 612*x^2 + 962*x - 347)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 - 56*x^17 + 324*x^16 - 356*x^15 - 1572*x^14 + 2528*x^13 + 2818*x^12 - 1914*x^11 - 7776*x^10 + 2496*x^9 + 4883*x^8 - 4924*x^7 - 4056*x^6 + 5314*x^5 + 2774*x^4 - 2944*x^3 + 612*x^2 + 962*x - 347, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} - 56 x^{17} + 324 x^{16} - 356 x^{15} - 1572 x^{14} + 2528 x^{13} + 2818 x^{12} - 1914 x^{11} - 7776 x^{10} + 2496 x^{9} + 4883 x^{8} - 4924 x^{7} - 4056 x^{6} + 5314 x^{5} + 2774 x^{4} - 2944 x^{3} + 612 x^{2} + 962 x - 347 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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}{418156295984677137369926895223487971108334729} a^{19} - \frac{156442614801603195834916818010204602503631299}{418156295984677137369926895223487971108334729} a^{18} + \frac{152602985319966762426694205853328537100902470}{418156295984677137369926895223487971108334729} a^{17} + \frac{122382131173016828906461503202255843882957555}{418156295984677137369926895223487971108334729} a^{16} - \frac{168463022659712166181378528280517937569149776}{418156295984677137369926895223487971108334729} a^{15} - \frac{28730043432631396157675433503955987619189242}{418156295984677137369926895223487971108334729} a^{14} + \frac{188659535481080018091662851726782499436922622}{418156295984677137369926895223487971108334729} a^{13} - \frac{125152113383071514537709551224044695505497015}{418156295984677137369926895223487971108334729} a^{12} - \frac{185689696922632967947145383470970373898019867}{418156295984677137369926895223487971108334729} a^{11} + \frac{59258946417785037627048811971005139666189927}{418156295984677137369926895223487971108334729} a^{10} - \frac{149146353546106637285031592747733981313334041}{418156295984677137369926895223487971108334729} a^{9} - \frac{180889328340577948831354630339776052325189697}{418156295984677137369926895223487971108334729} a^{8} - \frac{6110656338155564243118852946437711583472898}{418156295984677137369926895223487971108334729} a^{7} - \frac{146696621639851764521656994500290653648075244}{418156295984677137369926895223487971108334729} a^{6} - \frac{6183670434147157389909784552508666792194340}{418156295984677137369926895223487971108334729} a^{5} - \frac{146330877794181376402452662428918055475980249}{418156295984677137369926895223487971108334729} a^{4} - \frac{122631628527063205634651769077239528570385237}{418156295984677137369926895223487971108334729} a^{3} - \frac{6641677407928936832449453929902585721289775}{32165868921898241336148222709499074700641133} a^{2} + \frac{33778447215558379215847041442712478619606458}{418156295984677137369926895223487971108334729} a - \frac{124369068677760809317156850400400258963416991}{418156295984677137369926895223487971108334729}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 667852533.879 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 R $20$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$71$71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
71.10.9.7$x^{10} + 568$$10$$1$$9$$C_{10}$$[\ ]_{10}$