Properties

Label 23.23.611...809.1
Degree $23$
Signature $[23, 0]$
Discriminant $6.112\times 10^{36}$
Root discriminant $39.76$
Ramified prime $47$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{23}$ (as 23T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 22*x^21 + 21*x^20 + 210*x^19 - 190*x^18 - 1140*x^17 + 969*x^16 + 3876*x^15 - 3060*x^14 - 8568*x^13 + 6188*x^12 + 12376*x^11 - 8008*x^10 - 11440*x^9 + 6435*x^8 + 6435*x^7 - 3003*x^6 - 2002*x^5 + 715*x^4 + 286*x^3 - 66*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^23 - x^22 - 22*x^21 + 21*x^20 + 210*x^19 - 190*x^18 - 1140*x^17 + 969*x^16 + 3876*x^15 - 3060*x^14 - 8568*x^13 + 6188*x^12 + 12376*x^11 - 8008*x^10 - 11440*x^9 + 6435*x^8 + 6435*x^7 - 3003*x^6 - 2002*x^5 + 715*x^4 + 286*x^3 - 66*x^2 - 12*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -66, 286, 715, -2002, -3003, 6435, 6435, -11440, -8008, 12376, 6188, -8568, -3060, 3876, 969, -1140, -190, 210, 21, -22, -1, 1]);
 

\(x^{23} - x^{22} - 22 x^{21} + 21 x^{20} + 210 x^{19} - 190 x^{18} - 1140 x^{17} + 969 x^{16} + 3876 x^{15} - 3060 x^{14} - 8568 x^{13} + 6188 x^{12} + 12376 x^{11} - 8008 x^{10} - 11440 x^{9} + 6435 x^{8} + 6435 x^{7} - 3003 x^{6} - 2002 x^{5} + 715 x^{4} + 286 x^{3} - 66 x^{2} - 12 x + 1\)  Toggle raw display

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

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[23, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6111571184724799803076702357055363809\)\(\medspace = 47^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.76$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $47$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $23$
This field is Galois and abelian over $\Q$.
Conductor:  \(47\)
Dirichlet character group:    $\lbrace$$\chi_{47}(1,·)$, $\chi_{47}(2,·)$, $\chi_{47}(3,·)$, $\chi_{47}(4,·)$, $\chi_{47}(6,·)$, $\chi_{47}(7,·)$, $\chi_{47}(8,·)$, $\chi_{47}(9,·)$, $\chi_{47}(12,·)$, $\chi_{47}(14,·)$, $\chi_{47}(16,·)$, $\chi_{47}(17,·)$, $\chi_{47}(18,·)$, $\chi_{47}(21,·)$, $\chi_{47}(24,·)$, $\chi_{47}(25,·)$, $\chi_{47}(27,·)$, $\chi_{47}(28,·)$, $\chi_{47}(32,·)$, $\chi_{47}(34,·)$, $\chi_{47}(36,·)$, $\chi_{47}(37,·)$, $\chi_{47}(42,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $22$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 68215743661.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 68215743661.4 \cdot 1}{2\sqrt{6111571184724799803076702357055363809}}\approx 0.115735897897$ (assuming GRH)

Galois group

$C_{23}$ (as 23T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 23
The 23 conjugacy class representatives for $C_{23}$
Character table for $C_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ R $23$ $23$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
47Data not computed