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Properties

Label 20.0.11833780246...2912.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 3^{10}\cdot 11^{19}$
Root discriminant $28.42$
Ramified primes $2, 3, 11$
Class number $8$
Class group $[2, 4]$
Galois group $C_5\times D_4$ (as 20T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![463, 428, -849, -688, 4643, -7892, 11673, -12648, 11729, -8688, 5896, -3140, 1765, -606, 349, -126, 91, -32, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 32*x^17 + 91*x^16 - 126*x^15 + 349*x^14 - 606*x^13 + 1765*x^12 - 3140*x^11 + 5896*x^10 - 8688*x^9 + 11729*x^8 - 12648*x^7 + 11673*x^6 - 7892*x^5 + 4643*x^4 - 688*x^3 - 849*x^2 + 428*x + 463)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 32*x^17 + 91*x^16 - 126*x^15 + 349*x^14 - 606*x^13 + 1765*x^12 - 3140*x^11 + 5896*x^10 - 8688*x^9 + 11729*x^8 - 12648*x^7 + 11673*x^6 - 7892*x^5 + 4643*x^4 - 688*x^3 - 849*x^2 + 428*x + 463, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 32 x^{17} + 91 x^{16} - 126 x^{15} + 349 x^{14} - 606 x^{13} + 1765 x^{12} - 3140 x^{11} + 5896 x^{10} - 8688 x^{9} + 11729 x^{8} - 12648 x^{7} + 11673 x^{6} - 7892 x^{5} + 4643 x^{4} - 688 x^{3} - 849 x^{2} + 428 x + 463 \)

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:  \(118337802465720092063736102912=2^{15}\cdot 3^{10}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.42$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{398} a^{18} - \frac{21}{199} a^{17} + \frac{6}{199} a^{16} + \frac{39}{199} a^{15} - \frac{135}{398} a^{14} + \frac{115}{398} a^{13} - \frac{49}{199} a^{12} - \frac{64}{199} a^{11} - \frac{143}{398} a^{10} + \frac{10}{199} a^{9} - \frac{63}{398} a^{8} + \frac{193}{398} a^{7} - \frac{129}{398} a^{6} + \frac{39}{398} a^{5} - \frac{39}{398} a^{4} - \frac{99}{199} a^{3} - \frac{15}{398} a^{2} + \frac{77}{398} a - \frac{133}{398}$, $\frac{1}{694274620752943753130183059164526646} a^{19} - \frac{37214049614815177275424647350774}{347137310376471876565091529582263323} a^{18} - \frac{82489366436496672092962696814999730}{347137310376471876565091529582263323} a^{17} - \frac{173557692501317977329327978914070849}{694274620752943753130183059164526646} a^{16} - \frac{46261300185262460231698986859220141}{694274620752943753130183059164526646} a^{15} - \frac{172796119216921953340234194165558165}{347137310376471876565091529582263323} a^{14} - \frac{125744429049185323759790699652889928}{347137310376471876565091529582263323} a^{13} + \frac{131000519448751509235198870712569843}{347137310376471876565091529582263323} a^{12} - \frac{162813274426622882916958468713697395}{347137310376471876565091529582263323} a^{11} - \frac{5200613638907262828720742281450386}{347137310376471876565091529582263323} a^{10} - \frac{116851712017890869383331743331591603}{347137310376471876565091529582263323} a^{9} + \frac{70824053583115598329316699401158445}{347137310376471876565091529582263323} a^{8} - \frac{26529995322809597947535323086129148}{347137310376471876565091529582263323} a^{7} + \frac{23466256668591371369596575764417351}{694274620752943753130183059164526646} a^{6} + \frac{313283565735175272767354820404058917}{694274620752943753130183059164526646} a^{5} + \frac{53444541304024476610091936216190885}{694274620752943753130183059164526646} a^{4} - \frac{101194204579198045795145637758096307}{347137310376471876565091529582263323} a^{3} + \frac{39665731059330833616621063439779401}{694274620752943753130183059164526646} a^{2} - \frac{121258514532291615090628349184954588}{347137310376471876565091529582263323} a + \frac{326130228344692737315730132712961649}{694274620752943753130183059164526646}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 125582.779517 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times D_4$ (as 20T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 25 conjugacy class representatives for $C_5\times D_4$
Character table for $C_5\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), 4.0.95832.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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

Sibling fields

Galois closure: data not computed
Degree 20 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 R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed