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Properties

Label 20.0.73526643621...1888.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{34}\cdot 97^{5}\cdot 2657^{4}$
Root discriminant $49.35$
Ramified primes $2, 97, 2657$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![378944, -47936, 11056, -29968, 964740, -1102432, 1459328, -985804, 860486, -393960, 264779, -83518, 46637, -9716, 5074, -644, 400, -36, 27, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 27*x^18 - 36*x^17 + 400*x^16 - 644*x^15 + 5074*x^14 - 9716*x^13 + 46637*x^12 - 83518*x^11 + 264779*x^10 - 393960*x^9 + 860486*x^8 - 985804*x^7 + 1459328*x^6 - 1102432*x^5 + 964740*x^4 - 29968*x^3 + 11056*x^2 - 47936*x + 378944)
 
gp: K = bnfinit(x^20 - 2*x^19 + 27*x^18 - 36*x^17 + 400*x^16 - 644*x^15 + 5074*x^14 - 9716*x^13 + 46637*x^12 - 83518*x^11 + 264779*x^10 - 393960*x^9 + 860486*x^8 - 985804*x^7 + 1459328*x^6 - 1102432*x^5 + 964740*x^4 - 29968*x^3 + 11056*x^2 - 47936*x + 378944, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 27 x^{18} - 36 x^{17} + 400 x^{16} - 644 x^{15} + 5074 x^{14} - 9716 x^{13} + 46637 x^{12} - 83518 x^{11} + 264779 x^{10} - 393960 x^{9} + 860486 x^{8} - 985804 x^{7} + 1459328 x^{6} - 1102432 x^{5} + 964740 x^{4} - 29968 x^{3} + 11056 x^{2} - 47936 x + 378944 \)

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:  \(7352664362157151362137165437861888=2^{34}\cdot 97^{5}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.35$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{16} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{32} a^{18} - \frac{1}{32} a^{16} - \frac{1}{16} a^{15} - \frac{1}{8} a^{13} + \frac{1}{16} a^{12} + \frac{5}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{86190785813031472440324117320804440948674655323584} a^{19} + \frac{145936483545623624304644571367542962397265999679}{10773848226628934055040514665100555118584331915448} a^{18} + \frac{950532223147373697489667916879876288592410146027}{86190785813031472440324117320804440948674655323584} a^{17} - \frac{699093041212726794851045313569052061630597359043}{43095392906515736220162058660402220474337327661792} a^{16} - \frac{474638368442154960420573047959727543603153085027}{10773848226628934055040514665100555118584331915448} a^{15} - \frac{2097959101586531120117391271556916998001913647023}{21547696453257868110081029330201110237168663830896} a^{14} + \frac{1850460538697226939211038924731889092638296760333}{43095392906515736220162058660402220474337327661792} a^{13} - \frac{3862308055247472049305831470558583335478836938}{1346731028328616756880064333137569389823041489431} a^{12} + \frac{1567785157745651670078996350223943001010288596237}{86190785813031472440324117320804440948674655323584} a^{11} + \frac{1032610871971394482599727457821657782445042862733}{21547696453257868110081029330201110237168663830896} a^{10} - \frac{46455548128573944357871663445115165217870637317}{86190785813031472440324117320804440948674655323584} a^{9} + \frac{1151485505582304414208965881027377534846814185523}{43095392906515736220162058660402220474337327661792} a^{8} - \frac{312978524012533904575350912037985962076481861225}{43095392906515736220162058660402220474337327661792} a^{7} - \frac{819068765445796234767163681308319386792541842259}{10773848226628934055040514665100555118584331915448} a^{6} + \frac{2103259433585925799426125563634438224143146026333}{10773848226628934055040514665100555118584331915448} a^{5} + \frac{518262978044574953282601435360599530172321713425}{5386924113314467027520257332550277559292165957724} a^{4} + \frac{3472968416350039142957112367162774015805947750309}{21547696453257868110081029330201110237168663830896} a^{3} + \frac{524492325764006050374682059130901487666276729399}{10773848226628934055040514665100555118584331915448} a^{2} - \frac{341555852926080680997828988231481991456677249963}{5386924113314467027520257332550277559292165957724} a + \frac{113448766021290101658024186573702620750607856424}{1346731028328616756880064333137569389823041489431}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -1 \) (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:  \( 192406928.902 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.6208.2, 10.6.925322313728.1

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 24 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2657Data not computed