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Properties

Label 20.6.10929603344...9968.3
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 1583^{5}$
Root discriminant $25.23$
Ramified primes $2, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -76, 322, -832, 1314, -976, -714, 2848, -3199, 1240, 622, -740, 503, -576, 406, -188, 85, -28, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 28*x^17 + 85*x^16 - 188*x^15 + 406*x^14 - 576*x^13 + 503*x^12 - 740*x^11 + 622*x^10 + 1240*x^9 - 3199*x^8 + 2848*x^7 - 714*x^6 - 976*x^5 + 1314*x^4 - 832*x^3 + 322*x^2 - 76*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 28*x^17 + 85*x^16 - 188*x^15 + 406*x^14 - 576*x^13 + 503*x^12 - 740*x^11 + 622*x^10 + 1240*x^9 - 3199*x^8 + 2848*x^7 - 714*x^6 - 976*x^5 + 1314*x^4 - 832*x^3 + 322*x^2 - 76*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 28 x^{17} + 85 x^{16} - 188 x^{15} + 406 x^{14} - 576 x^{13} + 503 x^{12} - 740 x^{11} + 622 x^{10} + 1240 x^{9} - 3199 x^{8} + 2848 x^{7} - 714 x^{6} - 976 x^{5} + 1314 x^{4} - 832 x^{3} + 322 x^{2} - 76 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10929603344110461118702419968=-\,2^{40}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.23$
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:  $2, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4845172236622801054814853793877} a^{19} - \frac{955640147149796120470164953149}{4845172236622801054814853793877} a^{18} - \frac{69895781066886146068262293300}{4845172236622801054814853793877} a^{17} - \frac{199987572357987102596949385119}{4845172236622801054814853793877} a^{16} - \frac{213256222794803501724954638279}{4845172236622801054814853793877} a^{15} - \frac{1623739076548481671556241002626}{4845172236622801054814853793877} a^{14} + \frac{1154742635079413961217202524219}{4845172236622801054814853793877} a^{13} - \frac{567418035972286239675728686472}{4845172236622801054814853793877} a^{12} + \frac{820999376866360696462414040458}{4845172236622801054814853793877} a^{11} + \frac{66816645657378859282219748462}{4845172236622801054814853793877} a^{10} + \frac{777143010972825150381759875015}{4845172236622801054814853793877} a^{9} + \frac{1633875872498569301516494841519}{4845172236622801054814853793877} a^{8} - \frac{678941843303353692974637534263}{4845172236622801054814853793877} a^{7} - \frac{1525936583969112233314658436695}{4845172236622801054814853793877} a^{6} + \frac{553454528922439209321473818905}{4845172236622801054814853793877} a^{5} + \frac{893307189644413011544478802192}{4845172236622801054814853793877} a^{4} - \frac{1072900497644947858234544242203}{4845172236622801054814853793877} a^{3} + \frac{1225998019758954132527427063106}{4845172236622801054814853793877} a^{2} - \frac{5257813856739746228448994730}{32087233355117887780230819827} a + \frac{78112931181201288939938401573}{4845172236622801054814853793877}$

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:  $12$
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:  \( 1995930.82912 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.82112970752.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
1583Data not computed