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Properties

Label 20.0.128...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $1.280\times 10^{25}$
Root discriminant \(18.00\)
Ramified primes $2,5$
Class number $2$
Class group [2]
Galois group $F_5$ (as 20T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1)
 
gp: K = bnfinit(y^20 + 5*y^16 - 4*y^15 + 30*y^14 + 20*y^13 + 25*y^12 + 60*y^11 + 46*y^10 + 60*y^9 + 25*y^8 + 20*y^7 + 30*y^6 - 4*y^5 + 5*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1)
 

\( x^{20} + 5 x^{16} - 4 x^{15} + 30 x^{14} + 20 x^{13} + 25 x^{12} + 60 x^{11} + 46 x^{10} + 60 x^{9} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12800000000000000000000000\) \(\medspace = 2^{30}\cdot 5^{23}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(18.00\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{23/20}\approx 18.00364738986388$
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $20$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{12}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{8}a^{13}+\frac{1}{16}a^{11}+\frac{1}{16}a^{10}+\frac{3}{16}a^{9}-\frac{1}{16}a^{8}-\frac{3}{16}a^{7}+\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{5}{16}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}+\frac{1}{16}a+\frac{3}{16}$, $\frac{1}{32}a^{16}+\frac{1}{32}a^{14}+\frac{1}{16}a^{13}+\frac{1}{32}a^{12}+\frac{1}{16}a^{11}-\frac{3}{16}a^{9}-\frac{1}{4}a^{8}-\frac{1}{16}a^{7}+\frac{3}{16}a^{5}+\frac{15}{32}a^{4}-\frac{1}{16}a^{3}-\frac{1}{32}a^{2}-\frac{1}{4}a-\frac{1}{32}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{15}-\frac{3}{32}a^{13}+\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{3}{16}a^{9}+\frac{1}{8}a^{8}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{32}a^{5}+\frac{3}{8}a^{4}-\frac{1}{32}a^{3}-\frac{3}{8}a^{2}-\frac{3}{32}a+\frac{7}{16}$, $\frac{1}{5728}a^{18}-\frac{9}{2864}a^{17}-\frac{35}{5728}a^{16}-\frac{17}{1432}a^{15}-\frac{347}{5728}a^{14}+\frac{7}{179}a^{13}+\frac{13}{716}a^{12}+\frac{215}{2864}a^{11}-\frac{15}{179}a^{10}+\frac{197}{2864}a^{9}+\frac{119}{716}a^{8}+\frac{573}{2864}a^{7}-\frac{433}{5728}a^{6}+\frac{235}{1432}a^{5}-\frac{2853}{5728}a^{4}-\frac{1287}{2864}a^{3}+\frac{323}{5728}a^{2}+\frac{1423}{2864}a+\frac{291}{716}$, $\frac{1}{5728}a^{19}-\frac{1}{5728}a^{17}+\frac{9}{2864}a^{16}-\frac{139}{5728}a^{15}+\frac{2}{179}a^{14}-\frac{259}{2864}a^{13}-\frac{281}{2864}a^{12}-\frac{129}{2864}a^{11}+\frac{22}{179}a^{10}-\frac{95}{2864}a^{9}+\frac{185}{1432}a^{8}-\frac{211}{5728}a^{7}+\frac{83}{716}a^{6}-\frac{253}{5728}a^{5}+\frac{423}{2864}a^{4}+\frac{2321}{5728}a^{3}+\frac{375}{1432}a^{2}+\frac{85}{179}a+\frac{725}{2864}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{241}{1432}a^{19}+\frac{391}{5728}a^{18}+\frac{411}{5728}a^{17}-\frac{225}{2864}a^{16}+\frac{4633}{5728}a^{15}-\frac{461}{1432}a^{14}+\frac{29467}{5728}a^{13}+\frac{27125}{5728}a^{12}+\frac{11269}{1432}a^{11}+\frac{31583}{2864}a^{10}+\frac{1988}{179}a^{9}+\frac{43565}{2864}a^{8}+\frac{4675}{716}a^{7}+\frac{23651}{5728}a^{6}+\frac{16785}{5728}a^{5}-\frac{1739}{1432}a^{4}+\frac{5039}{5728}a^{3}-\frac{7987}{2864}a^{2}-\frac{6055}{5728}a-\frac{2663}{5728}$, $\frac{1563}{5728}a^{19}+\frac{697}{2864}a^{18}-\frac{879}{5728}a^{17}-\frac{787}{5728}a^{16}+\frac{7645}{5728}a^{15}+\frac{985}{5728}a^{14}+\frac{9279}{1432}a^{13}+\frac{72607}{5728}a^{12}+\frac{21471}{2864}a^{11}+\frac{44671}{2864}a^{10}+\frac{56323}{2864}a^{9}+\frac{44523}{2864}a^{8}+\frac{42655}{5728}a^{7}-\frac{6271}{1432}a^{6}+\frac{19921}{5728}a^{5}+\frac{8557}{5728}a^{4}-\frac{23083}{5728}a^{3}-\frac{1039}{5728}a^{2}-\frac{319}{2864}a+\frac{2499}{5728}$, $\frac{495}{5728}a^{19}+\frac{49}{716}a^{18}+\frac{63}{716}a^{17}-\frac{335}{5728}a^{16}+\frac{1047}{2864}a^{15}+\frac{13}{5728}a^{14}+\frac{15947}{5728}a^{13}+\frac{18011}{5728}a^{12}+\frac{8691}{1432}a^{11}+\frac{20553}{2864}a^{10}+\frac{2589}{358}a^{9}+\frac{31253}{2864}a^{8}+\frac{34401}{5728}a^{7}+\frac{8051}{2864}a^{6}+\frac{1249}{1432}a^{5}-\frac{4907}{5728}a^{4}+\frac{6309}{2864}a^{3}-\frac{10311}{5728}a^{2}-\frac{2619}{5728}a+\frac{1247}{5728}$, $\frac{1247}{5728}a^{19}-\frac{495}{5728}a^{18}-\frac{49}{716}a^{17}-\frac{63}{716}a^{16}+\frac{3285}{2864}a^{15}-\frac{3541}{2864}a^{14}+\frac{37397}{5728}a^{13}+\frac{8993}{5728}a^{12}+\frac{3291}{1432}a^{11}+\frac{5007}{716}a^{10}+\frac{508}{179}a^{9}+\frac{8349}{1432}a^{8}-\frac{31331}{5728}a^{7}-\frac{9461}{5728}a^{6}+\frac{5327}{1432}a^{5}-\frac{312}{179}a^{4}+\frac{5571}{2864}a^{3}-\frac{6309}{2864}a^{2}+\frac{10311}{5728}a+\frac{2619}{5728}$, $\frac{79}{179}a^{19}+\frac{245}{1432}a^{18}-\frac{299}{5728}a^{17}-\frac{63}{1432}a^{16}+\frac{12463}{5728}a^{15}-\frac{1249}{1432}a^{14}+\frac{70473}{5728}a^{13}+\frac{39945}{2864}a^{12}+\frac{36917}{2864}a^{11}+\frac{41107}{1432}a^{10}+\frac{78379}{2864}a^{9}+\frac{43811}{1432}a^{8}+\frac{46921}{2864}a^{7}+\frac{10337}{1432}a^{6}+\frac{75945}{5728}a^{5}+\frac{405}{358}a^{4}+\frac{5723}{5728}a^{3}-\frac{431}{716}a^{2}-\frac{5235}{5728}a-\frac{565}{2864}$, $\frac{317}{5728}a^{19}+\frac{615}{2864}a^{18}+\frac{69}{1432}a^{17}-\frac{649}{5728}a^{16}+\frac{339}{1432}a^{15}+\frac{4641}{5728}a^{14}+\frac{6241}{5728}a^{13}+\frac{39087}{5728}a^{12}+\frac{21287}{2864}a^{11}+\frac{4395}{716}a^{10}+\frac{38923}{2864}a^{9}+\frac{15939}{1432}a^{8}+\frac{53893}{5728}a^{7}+\frac{6135}{2864}a^{6}+\frac{15}{2864}a^{5}+\frac{14465}{5728}a^{4}-\frac{169}{2864}a^{3}+\frac{3387}{5728}a^{2}+\frac{5781}{5728}a-\frac{1835}{5728}$, $\frac{1351}{5728}a^{19}-\frac{861}{5728}a^{18}-\frac{889}{5728}a^{17}-\frac{321}{5728}a^{16}+\frac{7515}{5728}a^{15}-\frac{9643}{5728}a^{14}+\frac{19775}{2864}a^{13}+\frac{719}{1432}a^{12}-\frac{617}{716}a^{11}+\frac{7261}{1432}a^{10}+\frac{150}{179}a^{9}-\frac{493}{716}a^{8}-\frac{55283}{5728}a^{7}-\frac{25367}{5728}a^{6}+\frac{21665}{5728}a^{5}-\frac{23939}{5728}a^{4}-\frac{9523}{5728}a^{3}+\frac{3147}{5728}a^{2}+\frac{3915}{2864}a+\frac{45}{179}$, $\frac{337}{1432}a^{19}-\frac{325}{2864}a^{18}+\frac{1223}{5728}a^{17}+\frac{29}{1432}a^{16}+\frac{7009}{5728}a^{15}-\frac{2147}{1432}a^{14}+\frac{49167}{5728}a^{13}+\frac{809}{1432}a^{12}+\frac{29143}{2864}a^{11}+\frac{22941}{1432}a^{10}+\frac{31901}{2864}a^{9}+\frac{33641}{1432}a^{8}+\frac{32897}{2864}a^{7}+\frac{54841}{2864}a^{6}+\frac{82755}{5728}a^{5}+\frac{2977}{716}a^{4}+\frac{63141}{5728}a^{3}+\frac{653}{716}a^{2}+\frac{16099}{5728}a+\frac{981}{1432}$, $\frac{119}{5728}a^{19}+\frac{801}{5728}a^{18}+\frac{499}{5728}a^{17}+\frac{241}{5728}a^{16}+\frac{591}{5728}a^{15}+\frac{3539}{5728}a^{14}+\frac{9}{16}a^{13}+\frac{793}{179}a^{12}+\frac{1033}{179}a^{11}+\frac{5551}{716}a^{10}+\frac{9057}{716}a^{9}+\frac{19333}{1432}a^{8}+\frac{100941}{5728}a^{7}+\frac{62975}{5728}a^{6}+\frac{53373}{5728}a^{5}+\frac{54711}{5728}a^{4}+\frac{23041}{5728}a^{3}+\frac{15857}{5728}a^{2}+\frac{1407}{2864}a+\frac{77}{716}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 26927.8287731 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 26927.8287731 \cdot 2}{2\cdot\sqrt{12800000000000000000000000}}\cr\approx \mathstrut & 0.721763699564 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 5*x^16 - 4*x^15 + 30*x^14 + 20*x^13 + 25*x^12 + 60*x^11 + 46*x^10 + 60*x^9 + 25*x^8 + 20*x^7 + 30*x^6 - 4*x^5 + 5*x^4 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_5$ (as 20T5):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2, 5.1.200000.1 x5, 10.2.200000000000.1 x5

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 5 sibling: 5.1.200000.1
Degree 10 sibling: 10.2.200000000000.1
Minimal sibling: 5.1.200000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{5}$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.2.0.1}{2} }^{10}$ ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.6.3$x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$$2$$2$$6$$C_4$$[3]^{2}$
\(5\) Copy content Toggle raw display Deg $20$$20$$1$$23$