↑

Properties

Label 20.0.56477888187...9376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 47^{10}$
Root discriminant $19.39$
Ramified primes $2, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:D_5$ (as 20T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![115, 102, -402, -970, -133, 1286, 1552, 686, 565, 594, 614, 482, 360, -18, 78, 26, -21, 12, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 12*x^17 - 21*x^16 + 26*x^15 + 78*x^14 - 18*x^13 + 360*x^12 + 482*x^11 + 614*x^10 + 594*x^9 + 565*x^8 + 686*x^7 + 1552*x^6 + 1286*x^5 - 133*x^4 - 970*x^3 - 402*x^2 + 102*x + 115)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 + 12*x^17 - 21*x^16 + 26*x^15 + 78*x^14 - 18*x^13 + 360*x^12 + 482*x^11 + 614*x^10 + 594*x^9 + 565*x^8 + 686*x^7 + 1552*x^6 + 1286*x^5 - 133*x^4 - 970*x^3 - 402*x^2 + 102*x + 115, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} + 12 x^{17} - 21 x^{16} + 26 x^{15} + 78 x^{14} - 18 x^{13} + 360 x^{12} + 482 x^{11} + 614 x^{10} + 594 x^{9} + 565 x^{8} + 686 x^{7} + 1552 x^{6} + 1286 x^{5} - 133 x^{4} - 970 x^{3} - 402 x^{2} + 102 x + 115 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56477888187717354967269376=2^{30}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.39$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2563903924895116970931409875612325} a^{19} - \frac{132379729320789386496534704246941}{2563903924895116970931409875612325} a^{18} + \frac{828816902821125363162088601205472}{2563903924895116970931409875612325} a^{17} + \frac{493048114728708713252003993165104}{2563903924895116970931409875612325} a^{16} + \frac{1013824740875826906634778127321448}{2563903924895116970931409875612325} a^{15} + \frac{29719587674644909511199173897204}{2563903924895116970931409875612325} a^{14} - \frac{607138747268742118289791740594728}{2563903924895116970931409875612325} a^{13} - \frac{249048181518636327753516940122451}{2563903924895116970931409875612325} a^{12} - \frac{660265367780926980987563506165076}{2563903924895116970931409875612325} a^{11} + \frac{1179344421176984295671047277819371}{2563903924895116970931409875612325} a^{10} + \frac{46393061683889402538015727118109}{512780784979023394186281975122465} a^{9} - \frac{828993652299705588171763749344311}{2563903924895116970931409875612325} a^{8} - \frac{182561330928393597710207898255456}{2563903924895116970931409875612325} a^{7} + \frac{14856586522751907408195503303809}{512780784979023394186281975122465} a^{6} - \frac{441281770277512098312479763382178}{2563903924895116970931409875612325} a^{5} - \frac{1154093264497039475546827648308472}{2563903924895116970931409875612325} a^{4} - \frac{26402768420457419243116188238024}{102556156995804678837256395024493} a^{3} - \frac{109798361083209841690566389270524}{512780784979023394186281975122465} a^{2} - \frac{448132675960804830468010321996822}{2563903924895116970931409875612325} a + \frac{195202752458926118722610952102397}{512780784979023394186281975122465}$

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:  $9$
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:  \( 34812.3985182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.1.2209.1, 10.0.234849287168.1, 10.2.4996793344.1, 10.0.234849287168.2

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$