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Properties

Label 20.10.2115524933...2464.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{18}\cdot 13^{5}\cdot 107^{5}$
Root discriminant $32.83$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -210, 1294, -4256, 8319, -9272, 3300, 6180, -10887, 8490, -4068, 1200, 543, -1332, 882, -106, -153, 68, 1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + x^18 + 68*x^17 - 153*x^16 - 106*x^15 + 882*x^14 - 1332*x^13 + 543*x^12 + 1200*x^11 - 4068*x^10 + 8490*x^9 - 10887*x^8 + 6180*x^7 + 3300*x^6 - 9272*x^5 + 8319*x^4 - 4256*x^3 + 1294*x^2 - 210*x + 13)
 
gp: K = bnfinit(x^20 - 6*x^19 + x^18 + 68*x^17 - 153*x^16 - 106*x^15 + 882*x^14 - 1332*x^13 + 543*x^12 + 1200*x^11 - 4068*x^10 + 8490*x^9 - 10887*x^8 + 6180*x^7 + 3300*x^6 - 9272*x^5 + 8319*x^4 - 4256*x^3 + 1294*x^2 - 210*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + x^{18} + 68 x^{17} - 153 x^{16} - 106 x^{15} + 882 x^{14} - 1332 x^{13} + 543 x^{12} + 1200 x^{11} - 4068 x^{10} + 8490 x^{9} - 10887 x^{8} + 6180 x^{7} + 3300 x^{6} - 9272 x^{5} + 8319 x^{4} - 4256 x^{3} + 1294 x^{2} - 210 x + 13 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2115524933097244001040375742464=-\,2^{20}\cdot 3^{18}\cdot 13^{5}\cdot 107^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6945504956750909815909} a^{19} + \frac{2061848833016609754902}{6945504956750909815909} a^{18} + \frac{958662738330859453399}{6945504956750909815909} a^{17} - \frac{3029878738720933859315}{6945504956750909815909} a^{16} - \frac{2449249381990606198351}{6945504956750909815909} a^{15} - \frac{2676389702650661207296}{6945504956750909815909} a^{14} + \frac{3198808247882528096435}{6945504956750909815909} a^{13} + \frac{92390007718050432172}{6945504956750909815909} a^{12} + \frac{2763576335028839188223}{6945504956750909815909} a^{11} + \frac{2276632559172743322299}{6945504956750909815909} a^{10} - \frac{1778137237741274432001}{6945504956750909815909} a^{9} - \frac{2826723475075563548855}{6945504956750909815909} a^{8} - \frac{2506635051215221414477}{6945504956750909815909} a^{7} + \frac{1996015956184904030475}{6945504956750909815909} a^{6} + \frac{1051108638037599090109}{6945504956750909815909} a^{5} + \frac{1956444750068866824229}{6945504956750909815909} a^{4} + \frac{1353641663101747341208}{6945504956750909815909} a^{3} - \frac{677707684206511369170}{6945504956750909815909} a^{2} + \frac{2106179457907583012635}{6945504956750909815909} a + \frac{2100599380349186897780}{6945504956750909815909}$

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:  $14$
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:  \( 81444070.3764 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.12.28$x^{12} + 12 x^{11} - 3 x^{10} + 3 x^{9} + 3 x^{8} + 6 x^{7} + 12 x^{6} + 9 x^{5} + 9 x^{4} + 9 x + 9$$6$$2$$12$12T34$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$107$107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$