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Properties

Label 20.12.1214556712...8125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{4}\cdot 5^{15}\cdot 23^{8}\cdot 89^{4}$
Root discriminant $35.83$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -69, -3558, 5447, 11792, -2777, 2620, -121, -11863, -2198, 2754, 3182, 1417, -1212, -522, 83, 88, 15, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 17*x^18 + 15*x^17 + 88*x^16 + 83*x^15 - 522*x^14 - 1212*x^13 + 1417*x^12 + 3182*x^11 + 2754*x^10 - 2198*x^9 - 11863*x^8 - 121*x^7 + 2620*x^6 - 2777*x^5 + 11792*x^4 + 5447*x^3 - 3558*x^2 - 69*x + 41)
 
gp: K = bnfinit(x^20 - 2*x^19 - 17*x^18 + 15*x^17 + 88*x^16 + 83*x^15 - 522*x^14 - 1212*x^13 + 1417*x^12 + 3182*x^11 + 2754*x^10 - 2198*x^9 - 11863*x^8 - 121*x^7 + 2620*x^6 - 2777*x^5 + 11792*x^4 + 5447*x^3 - 3558*x^2 - 69*x + 41, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 17 x^{18} + 15 x^{17} + 88 x^{16} + 83 x^{15} - 522 x^{14} - 1212 x^{13} + 1417 x^{12} + 3182 x^{11} + 2754 x^{10} - 2198 x^{9} - 11863 x^{8} - 121 x^{7} + 2620 x^{6} - 2777 x^{5} + 11792 x^{4} + 5447 x^{3} - 3558 x^{2} - 69 x + 41 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12145567127297495495635986328125=3^{4}\cdot 5^{15}\cdot 23^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.83$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{12} + \frac{2}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{4}{27} a^{15} - \frac{4}{27} a^{14} + \frac{1}{3} a^{13} + \frac{4}{27} a^{12} + \frac{1}{27} a^{11} + \frac{2}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{27} a^{8} - \frac{4}{9} a^{7} + \frac{1}{27} a^{6} + \frac{7}{27} a^{5} + \frac{2}{27} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{10}{27}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} - \frac{8}{27} a^{13} - \frac{1}{27} a^{11} + \frac{4}{9} a^{10} + \frac{1}{27} a^{9} + \frac{8}{27} a^{8} - \frac{11}{27} a^{7} + \frac{4}{9} a^{6} + \frac{4}{27} a^{5} - \frac{11}{27} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{7}{27} a - \frac{5}{27}$, $\frac{1}{1233058499245084479934608938146536156321} a^{19} - \frac{26237868362001171419288012833768316}{411019499748361493311536312715512052107} a^{18} - \frac{19235954283109474916849747215573878890}{1233058499245084479934608938146536156321} a^{17} + \frac{67544448101133309557754756467843407646}{1233058499245084479934608938146536156321} a^{16} - \frac{199533369626945823044746841515446297553}{1233058499245084479934608938146536156321} a^{15} - \frac{50309682254838936606504848512270856510}{411019499748361493311536312715512052107} a^{14} - \frac{33834371503373844380937017974571931787}{137006499916120497770512104238504017369} a^{13} - \frac{38107315975643884102591882957118334221}{137006499916120497770512104238504017369} a^{12} - \frac{41750556429280363643313553843147333970}{1233058499245084479934608938146536156321} a^{11} + \frac{353783708009070989393254623542881250743}{1233058499245084479934608938146536156321} a^{10} - \frac{457146640907704194385662336655269172822}{1233058499245084479934608938146536156321} a^{9} - \frac{392918210906175533738472829052431160383}{1233058499245084479934608938146536156321} a^{8} + \frac{119550732916178986225007782577008183976}{411019499748361493311536312715512052107} a^{7} + \frac{453682827418973691048662380905024120509}{1233058499245084479934608938146536156321} a^{6} - \frac{137245810743982481452451289823435985599}{1233058499245084479934608938146536156321} a^{5} + \frac{248188660851172208924203661024903714297}{1233058499245084479934608938146536156321} a^{4} - \frac{119467518450102182324477702892372063527}{411019499748361493311536312715512052107} a^{3} - \frac{48148288779458152418537551392423693496}{1233058499245084479934608938146536156321} a^{2} + \frac{213847306046514780179851081512957285208}{1233058499245084479934608938146536156321} a - \frac{272393727261180408182942213363544641371}{1233058499245084479934608938146536156321}$

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:  $15$
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:  \( 262816482.728 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$