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Properties

Label 20.20.5168539800...0944.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{8}\cdot 13^{10}$
Root discriminant $86.23$
Ramified primes $2, 3, 7, 11, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_5^2$ (as 20T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15313, 67764, -65073, -635688, -423490, 1666554, 2411685, -714960, -3200232, -1615050, 616108, 735960, 77348, -103968, -27524, 5760, 2380, -108, -83, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313)
 
gp: K = bnfinit(x^20 - 83*x^18 - 108*x^17 + 2380*x^16 + 5760*x^15 - 27524*x^14 - 103968*x^13 + 77348*x^12 + 735960*x^11 + 616108*x^10 - 1615050*x^9 - 3200232*x^8 - 714960*x^7 + 2411685*x^6 + 1666554*x^5 - 423490*x^4 - 635688*x^3 - 65073*x^2 + 67764*x + 15313, 1)
 

Normalized defining polynomial

\( x^{20} - 83 x^{18} - 108 x^{17} + 2380 x^{16} + 5760 x^{15} - 27524 x^{14} - 103968 x^{13} + 77348 x^{12} + 735960 x^{11} + 616108 x^{10} - 1615050 x^{9} - 3200232 x^{8} - 714960 x^{7} + 2411685 x^{6} + 1666554 x^{5} - 423490 x^{4} - 635688 x^{3} - 65073 x^{2} + 67764 x + 15313 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(516853980088694624404163683037236690944=2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{8}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.23$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{4}{13} a^{14} + \frac{5}{13} a^{13} - \frac{1}{13} a^{12} + \frac{4}{13} a^{11} - \frac{2}{13} a^{10} - \frac{1}{13} a^{9} + \frac{2}{13} a^{8} - \frac{6}{13} a^{7} + \frac{6}{13} a^{6} - \frac{6}{13} a^{5} + \frac{6}{13} a^{4} - \frac{5}{13} a^{3} - \frac{4}{13} a^{2} - \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{17} + \frac{4}{13} a^{15} + \frac{5}{13} a^{14} - \frac{1}{13} a^{13} + \frac{4}{13} a^{12} - \frac{2}{13} a^{11} - \frac{1}{13} a^{10} + \frac{2}{13} a^{9} - \frac{6}{13} a^{8} + \frac{6}{13} a^{7} - \frac{6}{13} a^{6} + \frac{6}{13} a^{5} - \frac{5}{13} a^{4} - \frac{4}{13} a^{3} - \frac{4}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{46657} a^{18} - \frac{1465}{46657} a^{17} + \frac{1280}{46657} a^{16} - \frac{13122}{46657} a^{15} + \frac{10739}{46657} a^{14} - \frac{4878}{46657} a^{13} + \frac{3158}{46657} a^{12} + \frac{1845}{46657} a^{11} + \frac{18363}{46657} a^{10} + \frac{434}{3589} a^{9} + \frac{155}{46657} a^{8} + \frac{16815}{46657} a^{7} - \frac{2463}{46657} a^{6} - \frac{10874}{46657} a^{5} - \frac{13194}{46657} a^{4} - \frac{16202}{46657} a^{3} + \frac{3045}{46657} a^{2} - \frac{9260}{46657} a + \frac{12859}{46657}$, $\frac{1}{4938054874973657968018430322951439661} a^{19} + \frac{34122508535139623971776275875247}{4938054874973657968018430322951439661} a^{18} + \frac{167276376025805321371551776392496702}{4938054874973657968018430322951439661} a^{17} - \frac{109475722373952048649846957199102901}{4938054874973657968018430322951439661} a^{16} + \frac{1435090721138362844448775909258571687}{4938054874973657968018430322951439661} a^{15} + \frac{458460882376750745963573474567843393}{4938054874973657968018430322951439661} a^{14} - \frac{41853877883287845409581717769965441}{4938054874973657968018430322951439661} a^{13} + \frac{1258673070213887994607462880374400332}{4938054874973657968018430322951439661} a^{12} - \frac{758386664210926556909070064627885625}{4938054874973657968018430322951439661} a^{11} + \frac{134660759285777868943126564164360344}{4938054874973657968018430322951439661} a^{10} - \frac{537801097085166334859874970016715488}{4938054874973657968018430322951439661} a^{9} - \frac{1952214657477221470121535378318062869}{4938054874973657968018430322951439661} a^{8} - \frac{1010069098236664528229873437587448553}{4938054874973657968018430322951439661} a^{7} + \frac{4056032273123034811562044653461787}{379850374997973689847571563303956897} a^{6} + \frac{1513926540270078002812618523133118856}{4938054874973657968018430322951439661} a^{5} + \frac{308336248766799185824237608355793003}{4938054874973657968018430322951439661} a^{4} + \frac{1101283140829825717284038232879137476}{4938054874973657968018430322951439661} a^{3} + \frac{908781023838014714521775050798018331}{4938054874973657968018430322951439661} a^{2} + \frac{2411308790992423517661632776620140938}{4938054874973657968018430322951439661} a - \frac{123977261531209698230593868475854162}{4938054874973657968018430322951439661}$

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:  $19$
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:  \( 8904025444480 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2$ (as 20T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{91}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{3}, \sqrt{91})\), 10.10.249828821987576832.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$