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Properties

Label 20.2.58313562010...4224.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $15.43$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 13, -70, 212, -431, 669, -822, 808, -606, 294, 5, -190, 268, -255, 217, -168, 112, -61, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 25*x^18 - 61*x^17 + 112*x^16 - 168*x^15 + 217*x^14 - 255*x^13 + 268*x^12 - 190*x^11 + 5*x^10 + 294*x^9 - 606*x^8 + 808*x^7 - 822*x^6 + 669*x^5 - 431*x^4 + 212*x^3 - 70*x^2 + 13*x - 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 25*x^18 - 61*x^17 + 112*x^16 - 168*x^15 + 217*x^14 - 255*x^13 + 268*x^12 - 190*x^11 + 5*x^10 + 294*x^9 - 606*x^8 + 808*x^7 - 822*x^6 + 669*x^5 - 431*x^4 + 212*x^3 - 70*x^2 + 13*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 25 x^{18} - 61 x^{17} + 112 x^{16} - 168 x^{15} + 217 x^{14} - 255 x^{13} + 268 x^{12} - 190 x^{11} + 5 x^{10} + 294 x^{9} - 606 x^{8} + 808 x^{7} - 822 x^{6} + 669 x^{5} - 431 x^{4} + 212 x^{3} - 70 x^{2} + 13 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-583135620108095986484224=-\,2^{10}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.43$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{77} a^{16} + \frac{20}{77} a^{15} + \frac{16}{77} a^{14} - \frac{8}{77} a^{13} + \frac{25}{77} a^{12} + \frac{4}{11} a^{11} - \frac{5}{77} a^{10} + \frac{16}{77} a^{9} - \frac{2}{11} a^{8} + \frac{6}{77} a^{7} - \frac{34}{77} a^{6} - \frac{30}{77} a^{5} - \frac{29}{77} a^{4} - \frac{20}{77} a^{3} + \frac{3}{7} a^{2} - \frac{10}{77} a - \frac{4}{77}$, $\frac{1}{77} a^{17} + \frac{1}{77} a^{15} - \frac{20}{77} a^{14} + \frac{31}{77} a^{13} - \frac{10}{77} a^{12} - \frac{26}{77} a^{11} - \frac{38}{77} a^{10} - \frac{26}{77} a^{9} - \frac{2}{7} a^{8} + \frac{34}{77} a^{6} + \frac{32}{77} a^{5} + \frac{3}{11} a^{4} - \frac{29}{77} a^{3} + \frac{23}{77} a^{2} - \frac{5}{11} a + \frac{3}{77}$, $\frac{1}{847} a^{18} + \frac{1}{847} a^{17} + \frac{5}{847} a^{16} - \frac{170}{847} a^{15} + \frac{306}{847} a^{14} + \frac{13}{77} a^{13} + \frac{295}{847} a^{12} + \frac{279}{847} a^{11} - \frac{45}{121} a^{10} + \frac{247}{847} a^{9} + \frac{153}{847} a^{8} + \frac{289}{847} a^{7} + \frac{12}{121} a^{6} + \frac{164}{847} a^{5} - \frac{47}{847} a^{4} + \frac{299}{847} a^{3} - \frac{111}{847} a^{2} + \frac{390}{847} a - \frac{13}{847}$, $\frac{1}{208998097} a^{19} + \frac{573}{29856871} a^{18} + \frac{2679}{1727257} a^{17} + \frac{119023}{208998097} a^{16} - \frac{45864598}{208998097} a^{15} + \frac{71613200}{208998097} a^{14} - \frac{9402046}{29856871} a^{13} + \frac{24788745}{208998097} a^{12} - \frac{86473073}{208998097} a^{11} - \frac{24139757}{208998097} a^{10} + \frac{59546703}{208998097} a^{9} + \frac{20727396}{208998097} a^{8} + \frac{35798150}{208998097} a^{7} - \frac{82527250}{208998097} a^{6} + \frac{37372172}{208998097} a^{5} - \frac{167040}{208998097} a^{4} - \frac{3163302}{18999827} a^{3} - \frac{91335344}{208998097} a^{2} - \frac{85675089}{208998097} a + \frac{36825161}{208998097}$

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:  $10$
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:  \( 5288.64508061 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.3409076657.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$