↑

Properties

Label 20.0.31551644091...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 251^{2}\cdot 4051^{2}$
Root discriminant $133.34$
Ramified primes $2, 5, 251, 4051$
Class number $39812704$ (GRH)
Class group $[2, 2, 2, 2, 2488294]$ (GRH)
Galois group 20T344

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5169421368005, 0, 9384437729375, 0, 3383696380375, 0, 566304002475, 0, 54008113350, 0, 3179289615, 0, 118780475, 0, 2798650, 0, 39870, 0, 310, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 310*x^18 + 39870*x^16 + 2798650*x^14 + 118780475*x^12 + 3179289615*x^10 + 54008113350*x^8 + 566304002475*x^6 + 3383696380375*x^4 + 9384437729375*x^2 + 5169421368005)
 
gp: K = bnfinit(x^20 + 310*x^18 + 39870*x^16 + 2798650*x^14 + 118780475*x^12 + 3179289615*x^10 + 54008113350*x^8 + 566304002475*x^6 + 3383696380375*x^4 + 9384437729375*x^2 + 5169421368005, 1)
 

Normalized defining polynomial

\( x^{20} + 310 x^{18} + 39870 x^{16} + 2798650 x^{14} + 118780475 x^{12} + 3179289615 x^{10} + 54008113350 x^{8} + 566304002475 x^{6} + 3383696380375 x^{4} + 9384437729375 x^{2} + 5169421368005 \)

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:  \(3155164409182739257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 251^{2}\cdot 4051^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.34$
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:  $2, 5, 251, 4051$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4724921148910191724856811889904322882588458104744249} a^{18} - \frac{1175640049967488918423911378666606292406455513575950}{4724921148910191724856811889904322882588458104744249} a^{16} + \frac{2260055934161755388312315568946778950160944047771613}{4724921148910191724856811889904322882588458104744249} a^{14} + \frac{6685817134401795690667976713484337007708866717026}{18824387047450963047238294382088935787205012369499} a^{12} + \frac{2296741715748550706223967057789653365214954626616166}{4724921148910191724856811889904322882588458104744249} a^{10} - \frac{1549467128189680429395659064539896443631778488666097}{4724921148910191724856811889904322882588458104744249} a^{8} - \frac{1817910854291680577146636021374454658749457899316776}{4724921148910191724856811889904322882588458104744249} a^{6} - \frac{57223247103868890360656175446089410705166629394}{4724921148910191724856811889904322882588458104744249} a^{4} - \frac{1992092639468369374519025356086042626320189709886032}{4724921148910191724856811889904322882588458104744249} a^{2} + \frac{1694519691879463202826456038638807340706812973}{4646849431609716871695456524830643245422121049}$, $\frac{1}{4724921148910191724856811889904322882588458104744249} a^{19} - \frac{1175640049967488918423911378666606292406455513575950}{4724921148910191724856811889904322882588458104744249} a^{17} + \frac{2260055934161755388312315568946778950160944047771613}{4724921148910191724856811889904322882588458104744249} a^{15} + \frac{6685817134401795690667976713484337007708866717026}{18824387047450963047238294382088935787205012369499} a^{13} + \frac{2296741715748550706223967057789653365214954626616166}{4724921148910191724856811889904322882588458104744249} a^{11} - \frac{1549467128189680429395659064539896443631778488666097}{4724921148910191724856811889904322882588458104744249} a^{9} - \frac{1817910854291680577146636021374454658749457899316776}{4724921148910191724856811889904322882588458104744249} a^{7} - \frac{57223247103868890360656175446089410705166629394}{4724921148910191724856811889904322882588458104744249} a^{5} - \frac{1992092639468369374519025356086042626320189709886032}{4724921148910191724856811889904322882588458104744249} a^{3} + \frac{1694519691879463202826456038638807340706812973}{4646849431609716871695456524830643245422121049} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2488294}$, which has order $39812704$ (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:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T344:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n344 are not computed
Character table for t20n344 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
251Data not computed
4051Data not computed