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Properties

Label 20.12.1053743761...9552.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 13^{7}\cdot 101^{10}\cdot 347^{4}$
Root discriminant $158.90$
Ramified primes $2, 13, 101, 347$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3007693, 0, -636927538, 0, 1098047444, 0, -796090326, 0, 269167585, 0, -46066504, 0, 4407877, 0, -247742, 0, 8100, 0, -141, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 141*x^18 + 8100*x^16 - 247742*x^14 + 4407877*x^12 - 46066504*x^10 + 269167585*x^8 - 796090326*x^6 + 1098047444*x^4 - 636927538*x^2 + 3007693)
 
gp: K = bnfinit(x^20 - 141*x^18 + 8100*x^16 - 247742*x^14 + 4407877*x^12 - 46066504*x^10 + 269167585*x^8 - 796090326*x^6 + 1098047444*x^4 - 636927538*x^2 + 3007693, 1)
 

Normalized defining polynomial

\( x^{20} - 141 x^{18} + 8100 x^{16} - 247742 x^{14} + 4407877 x^{12} - 46066504 x^{10} + 269167585 x^{8} - 796090326 x^{6} + 1098047444 x^{4} - 636927538 x^{2} + 3007693 \)

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:  \(105374376135245356088725314843202362306199552=2^{20}\cdot 13^{7}\cdot 101^{10}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.90$
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:  $2, 13, 101, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{2}{13} a^{14} + \frac{1}{13} a^{12} - \frac{1}{13} a^{10} + \frac{6}{13} a^{8} - \frac{3}{13} a^{6} - \frac{2}{13} a^{4} - \frac{5}{13} a^{2}$, $\frac{1}{13} a^{17} + \frac{2}{13} a^{15} + \frac{1}{13} a^{13} - \frac{1}{13} a^{11} + \frac{6}{13} a^{9} - \frac{3}{13} a^{7} - \frac{2}{13} a^{5} - \frac{5}{13} a^{3}$, $\frac{1}{42258650214251056541500144059803677594949} a^{18} + \frac{302410721047550689162926798324625819280}{42258650214251056541500144059803677594949} a^{16} - \frac{9929124087645507978747694102173543172542}{42258650214251056541500144059803677594949} a^{14} + \frac{1933437818075762088819669724205688319832}{42258650214251056541500144059803677594949} a^{12} - \frac{11709505445629777711363079535280467628048}{42258650214251056541500144059803677594949} a^{10} + \frac{17691479376215933524367899259286498115526}{42258650214251056541500144059803677594949} a^{8} + \frac{5111041059159265219366041759066505594865}{42258650214251056541500144059803677594949} a^{6} - \frac{19859405162776814124378166574216879409794}{42258650214251056541500144059803677594949} a^{4} - \frac{159530128743493786806012423223844095852}{3250665401096235118576934158446436738073} a^{2} + \frac{24049784461002716513599457746450673040}{250051184699710393736687242957418210621}$, $\frac{1}{1563570057927289092035505330212736071013113} a^{19} - \frac{2948254680048684429414007360121810918793}{1563570057927289092035505330212736071013113} a^{17} + \frac{575190648109676813365100454418185069680598}{1563570057927289092035505330212736071013113} a^{15} - \frac{677455631011037377693759569391099589937425}{1563570057927289092035505330212736071013113} a^{13} + \frac{202834411026721740114714574922184357084770}{1563570057927289092035505330212736071013113} a^{11} - \frac{44071163244612533728593849751195799907861}{1563570057927289092035505330212736071013113} a^{9} + \frac{606484140261962762156098861071657302138370}{1563570057927289092035505330212736071013113} a^{7} - \frac{689496477788601248551226603214182847452832}{1563570057927289092035505330212736071013113} a^{5} + \frac{40098710607909879604800633692920487814129}{120274619840560699387346563862518159308701} a^{3} + \frac{2024459262058685866407097401405796358008}{9251893833889284568257427989424473792977} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 647131009893000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 5.3.4511.1, 10.6.213871306817009621.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101Data not computed
347Data not computed