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Properties

Label 20.10.1825672741...6811.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 331^{5}$
Root discriminant $29.04$
Ramified primes $11, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -119, -833, -71, 5979, 3150, -2075, 452, 2581, 3176, -2426, -1179, 2149, -654, -387, 372, -105, -17, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 21*x^18 - 17*x^17 - 105*x^16 + 372*x^15 - 387*x^14 - 654*x^13 + 2149*x^12 - 1179*x^11 - 2426*x^10 + 3176*x^9 + 2581*x^8 + 452*x^7 - 2075*x^6 + 3150*x^5 + 5979*x^4 - 71*x^3 - 833*x^2 - 119*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 21*x^18 - 17*x^17 - 105*x^16 + 372*x^15 - 387*x^14 - 654*x^13 + 2149*x^12 - 1179*x^11 - 2426*x^10 + 3176*x^9 + 2581*x^8 + 452*x^7 - 2075*x^6 + 3150*x^5 + 5979*x^4 - 71*x^3 - 833*x^2 - 119*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 21 x^{18} - 17 x^{17} - 105 x^{16} + 372 x^{15} - 387 x^{14} - 654 x^{13} + 2149 x^{12} - 1179 x^{11} - 2426 x^{10} + 3176 x^{9} + 2581 x^{8} + 452 x^{7} - 2075 x^{6} + 3150 x^{5} + 5979 x^{4} - 71 x^{3} - 833 x^{2} - 119 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-182567274194490055859030886811=-\,11^{16}\cdot 331^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.04$
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:  $11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{126245586568171315755459655649836109183} a^{19} - \frac{18389700637384181945036338433621044442}{126245586568171315755459655649836109183} a^{18} + \frac{24263133141364729072586052936419064294}{126245586568171315755459655649836109183} a^{17} - \frac{42422889809956232474167608057832889080}{126245586568171315755459655649836109183} a^{16} + \frac{60262126061683233961726584576957882812}{126245586568171315755459655649836109183} a^{15} - \frac{56230761763656736906563751095839217138}{126245586568171315755459655649836109183} a^{14} - \frac{49467519622089706009715856755287938626}{126245586568171315755459655649836109183} a^{13} + \frac{280312603038263152888703208707932862}{640840540955184343936343429694599539} a^{12} - \frac{26446161077486629245519264230090555728}{126245586568171315755459655649836109183} a^{11} + \frac{3321307454566991881555789994750611642}{126245586568171315755459655649836109183} a^{10} - \frac{50812213817669748940068204835724648180}{126245586568171315755459655649836109183} a^{9} + \frac{38287384509838606803525963387045463264}{126245586568171315755459655649836109183} a^{8} - \frac{24100255985647248227841898207526165110}{126245586568171315755459655649836109183} a^{7} - \frac{10492929448846445403002159550064100483}{126245586568171315755459655649836109183} a^{6} + \frac{12541490444960409025963858926630906228}{126245586568171315755459655649836109183} a^{5} - \frac{5439659665185021661700610362913777615}{126245586568171315755459655649836109183} a^{4} - \frac{15715604265943180935926324973391224664}{126245586568171315755459655649836109183} a^{3} + \frac{44261670051536718752606473415616210585}{126245586568171315755459655649836109183} a^{2} - \frac{53883409161238986284588795349590789928}{126245586568171315755459655649836109183} a + \frac{14369840718015097609127898510153323595}{126245586568171315755459655649836109183}$

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:  $14$
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:  \( 15014513.7978 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.70952789611.1

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
Arithmetically equvalently 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 $20$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
11Data not computed
331Data not computed