Properties

Label 20.0.30206313154...2896.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 220873^{2}$
Root discriminant $11.86$
Ramified primes $2, 3, 220873$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, -1, -2, 0, -2, 4, -6, 7, -6, 8, -6, 7, -4, 2, -4, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 1)
gp: K = bnfinit(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut +\mathstrut x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 2 x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 7 x^{14} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut +\mathstrut 8 x^{12} \) \(\mathstrut -\mathstrut 6 x^{11} \) \(\mathstrut +\mathstrut 7 x^{10} \) \(\mathstrut -\mathstrut 6 x^{9} \) \(\mathstrut +\mathstrut 4 x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(3020631315406201552896=2^{20}\cdot 3^{10}\cdot 220873^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.86$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 220873$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{1745} a^{19} + \frac{12}{349} a^{18} - \frac{238}{1745} a^{17} + \frac{723}{1745} a^{16} - \frac{592}{1745} a^{15} - \frac{624}{1745} a^{14} - \frac{18}{349} a^{13} - \frac{869}{1745} a^{12} + \frac{567}{1745} a^{11} - \frac{537}{1745} a^{10} - \frac{803}{1745} a^{9} - \frac{373}{1745} a^{8} - \frac{738}{1745} a^{7} + \frac{41}{1745} a^{6} - \frac{332}{1745} a^{5} + \frac{64}{349} a^{4} - \frac{694}{1745} a^{3} + \frac{589}{1745} a^{2} + \frac{158}{349} a - \frac{413}{1745}$

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

Class group and class number

Trivial Abelian group, 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:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{57}{349} a^{19} - \frac{70}{349} a^{18} + \frac{45}{349} a^{17} - \frac{320}{349} a^{16} + \frac{458}{349} a^{15} - \frac{319}{349} a^{14} + \frac{803}{349} a^{13} - \frac{1022}{349} a^{12} + \frac{909}{349} a^{11} - \frac{944}{349} a^{10} + \frac{995}{349} a^{9} - \frac{1019}{349} a^{8} + \frac{512}{349} a^{7} - \frac{106}{349} a^{6} + \frac{271}{349} a^{5} + \frac{92}{349} a^{4} - \frac{121}{349} a^{3} + \frac{418}{349} a^{2} + \frac{9}{349} a - \frac{158}{349} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{158}{349} a^{19} + \frac{57}{349} a^{18} + \frac{88}{349} a^{17} - \frac{587}{349} a^{16} - \frac{4}{349} a^{15} - \frac{174}{349} a^{14} + \frac{787}{349} a^{13} - \frac{145}{349} a^{12} + \frac{242}{349} a^{11} - \frac{39}{349} a^{10} + \frac{162}{349} a^{9} + \frac{47}{349} a^{8} - \frac{387}{349} a^{7} + \frac{196}{349} a^{6} - \frac{106}{349} a^{5} - \frac{45}{349} a^{4} - \frac{66}{349} a^{3} - \frac{121}{349} a^{2} + \frac{576}{349} a + \frac{9}{349} \),  \( \frac{1416}{1745} a^{19} + \frac{153}{1745} a^{18} + \frac{1173}{1745} a^{17} - \frac{5084}{1745} a^{16} + \frac{2818}{1745} a^{15} - \frac{751}{349} a^{14} + \frac{8321}{1745} a^{13} - \frac{7957}{1745} a^{12} + \frac{8199}{1745} a^{11} - \frac{6552}{1745} a^{10} + \frac{8021}{1745} a^{9} - \frac{6064}{1745} a^{8} + \frac{5831}{1745} a^{7} - \frac{185}{349} a^{6} + \frac{689}{1745} a^{5} - \frac{2674}{1745} a^{4} + \frac{778}{1745} a^{3} - \frac{784}{1745} a^{2} + \frac{1491}{1745} a - \frac{931}{1745} \),  \( \frac{588}{1745} a^{19} + \frac{1078}{1745} a^{18} + \frac{1052}{1745} a^{17} - \frac{1703}{1745} a^{16} - \frac{2586}{1745} a^{15} - \frac{1858}{1745} a^{14} + \frac{2571}{1745} a^{13} + \frac{272}{349} a^{12} + \frac{1148}{1745} a^{11} + \frac{89}{1745} a^{10} + \frac{565}{349} a^{9} + \frac{179}{349} a^{8} - \frac{167}{349} a^{7} + \frac{27}{1745} a^{6} - \frac{25}{349} a^{5} - \frac{649}{1745} a^{4} - \frac{437}{349} a^{3} + \frac{124}{1745} a^{2} + \frac{1}{1745} a + \frac{758}{1745} \),  \( \frac{1114}{1745} a^{19} + \frac{879}{1745} a^{18} + \frac{806}{1745} a^{17} - \frac{3909}{1745} a^{16} - \frac{1623}{1745} a^{15} - \frac{1324}{1745} a^{14} + \frac{5138}{1745} a^{13} + \frac{12}{349} a^{12} + \frac{1344}{1745} a^{11} + \frac{317}{1745} a^{10} + \frac{338}{349} a^{9} + \frac{167}{349} a^{8} - \frac{536}{349} a^{7} + \frac{3096}{1745} a^{6} - \frac{540}{349} a^{5} - \frac{547}{1745} a^{4} - \frac{784}{349} a^{3} + \frac{1422}{1745} a^{2} + \frac{1278}{1745} a + \frac{249}{1745} \),  \( \frac{427}{1745} a^{19} - \frac{1253}{1745} a^{18} - \frac{67}{1745} a^{17} - \frac{2587}{1745} a^{16} + \frac{5476}{1745} a^{15} - \frac{1557}{1745} a^{14} + \frac{5544}{1745} a^{13} - \frac{1830}{349} a^{12} + \frac{5487}{1745} a^{11} - \frac{5939}{1745} a^{10} + \frac{1154}{349} a^{9} - \frac{1212}{349} a^{8} + \frac{423}{349} a^{7} - \frac{292}{1745} a^{6} - \frac{14}{349} a^{5} + \frac{879}{1745} a^{4} + \frac{202}{349} a^{3} + \frac{2666}{1745} a^{2} + \frac{894}{1745} a - \frac{2898}{1745} \),  \( \frac{1098}{1745} a^{19} - \frac{86}{349} a^{18} + \frac{426}{1745} a^{17} - \frac{5356}{1745} a^{16} + \frac{2614}{1745} a^{15} - \frac{2857}{1745} a^{14} + \frac{1874}{349} a^{13} - \frac{6627}{1745} a^{12} + \frac{6581}{1745} a^{11} - \frac{6796}{1745} a^{10} + \frac{6511}{1745} a^{9} - \frac{6459}{1745} a^{8} + \frac{1101}{1745} a^{7} - \frac{2097}{1745} a^{6} - \frac{1576}{1745} a^{5} - \frac{575}{349} a^{4} - \frac{1192}{1745} a^{3} + \frac{2817}{1745} a^{2} + \frac{729}{349} a + \frac{226}{1745} \),  \( \frac{1627}{1745} a^{19} + \frac{1296}{1745} a^{18} + \frac{1211}{1745} a^{17} - \frac{5393}{1745} a^{16} - \frac{1689}{1745} a^{15} - \frac{490}{349} a^{14} + \frac{6432}{1745} a^{13} - \frac{1809}{1745} a^{12} + \frac{3243}{1745} a^{11} + \frac{546}{1745} a^{10} + \frac{4712}{1745} a^{9} - \frac{658}{1745} a^{8} - \frac{1213}{1745} a^{7} + \frac{219}{349} a^{6} - \frac{1657}{1745} a^{5} - \frac{3558}{1745} a^{4} - \frac{3264}{1745} a^{3} - \frac{1098}{1745} a^{2} + \frac{2057}{1745} a + \frac{223}{1745} \),  \( \frac{177}{349} a^{19} + \frac{401}{1745} a^{18} + \frac{1562}{1745} a^{17} - \frac{2654}{1745} a^{16} + \frac{265}{349} a^{15} - \frac{3612}{1745} a^{14} + \frac{5157}{1745} a^{13} - \frac{4406}{1745} a^{12} + \frac{4819}{1745} a^{11} - \frac{819}{349} a^{10} + \frac{5493}{1745} a^{9} - \frac{1347}{1745} a^{8} + \frac{3688}{1745} a^{7} + \frac{2083}{1745} a^{6} + \frac{2132}{1745} a^{5} - \frac{188}{1745} a^{4} - \frac{1346}{1745} a^{3} - \frac{1886}{1745} a^{2} + \frac{452}{1745} a - \frac{2196}{1745} \),  \( \frac{529}{1745} a^{19} + \frac{679}{1745} a^{18} + \frac{436}{1745} a^{17} - \frac{1084}{1745} a^{16} - \frac{813}{1745} a^{15} + \frac{756}{1745} a^{14} + \frac{203}{1745} a^{13} + \frac{126}{349} a^{12} - \frac{2291}{1745} a^{11} + \frac{3852}{1745} a^{10} - \frac{290}{349} a^{9} + \frac{881}{349} a^{8} - \frac{393}{349} a^{7} + \frac{5286}{1745} a^{6} - \frac{435}{349} a^{5} + \frac{713}{1745} a^{4} - \frac{205}{349} a^{3} + \frac{622}{1745} a^{2} - \frac{192}{1745} a - \frac{701}{1745} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1171.34610492 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T656:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 10.0.226173952.1

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 24 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
220873Data not computed