magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121104, -7656, 242, -1590, 777, 14, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 8*x^6 + 14*x^5 + 777*x^4 - 1590*x^3 + 242*x^2 - 7656*x + 121104)
gp: K = bnfinit(x^8 - 4*x^7 + 8*x^6 + 14*x^5 + 777*x^4 - 1590*x^3 + 242*x^2 - 7656*x + 121104, 1)
Normalized defining polynomial
\( x^{8} - 4 x^{7} + 8 x^{6} + 14 x^{5} + 777 x^{4} - 1590 x^{3} + 242 x^{2} - 7656 x + 121104 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1341515445903616=2^{8}\cdot 17^{4}\cdot 89^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $77.79$ | 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: | $2, 17, 89$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{444} a^{6} - \frac{1}{148} a^{5} + \frac{23}{444} a^{4} - \frac{17}{444} a^{3} - \frac{79}{222} a^{2} + \frac{83}{222} a - \frac{18}{37}$, $\frac{1}{9554352528} a^{7} - \frac{2469013}{3184784176} a^{6} + \frac{199863977}{9554352528} a^{5} - \frac{1159660889}{9554352528} a^{4} + \frac{510179969}{1194294066} a^{3} + \frac{427257539}{4777176264} a^{2} - \frac{69440617}{199049011} a - \frac{951611}{6863759}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{6}\times C_{12}$, which has order $144$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $3$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{2053}{86075248} a^{7} - \frac{71761}{258225744} a^{6} + \frac{37565}{86075248} a^{5} + \frac{513193}{258225744} a^{4} - \frac{17981}{32278218} a^{3} - \frac{13299155}{129112872} a^{2} - \frac{494512}{16139109} a + \frac{121516}{185507} \) (order $4$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3676.45676734 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1513}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{1513}) \), \(\Q(i, \sqrt{1513})\), 4.0.24208.1 x2, 4.2.9156676.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.9156676.1, 4.0.24208.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $89$ | 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 89.4.2.1 | $x^{4} + 979 x^{2} + 285156$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.17_89.2t1.1c1 | $1$ | $ 17 \cdot 89 $ | $x^{2} - x - 378$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.2e2_17_89.2t1.1c1 | $1$ | $ 2^{2} \cdot 17 \cdot 89 $ | $x^{2} + 1513$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.2e2_17_89.4t3.17c1 | $2$ | $ 2^{2} \cdot 17 \cdot 89 $ | $x^{8} - 4 x^{7} + 8 x^{6} + 14 x^{5} + 777 x^{4} - 1590 x^{3} + 242 x^{2} - 7656 x + 121104$ | $D_4$ (as 8T4) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.