Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 3*x^11 - 3*x^10 + 5*x^9 + 2*x^8 - 17*x^7 + 10*x^6 + 16*x^5 - 16*x^4 - 5*x^3 + 7*x^2 - 1)
gp: K = bnfinit(x^13 - 3*x^12 + 3*x^11 - 3*x^10 + 5*x^9 + 2*x^8 - 17*x^7 + 10*x^6 + 16*x^5 - 16*x^4 - 5*x^3 + 7*x^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 7, -5, -16, 16, 10, -17, 2, 5, -3, 3, -3, 1]);
\(x^{13} - 3 x^{12} + 3 x^{11} - 3 x^{10} + 5 x^{9} + 2 x^{8} - 17 x^{7} + 10 x^{6} + 16 x^{5} - 16 x^{4} - 5 x^{3} + 7 x^{2} - 1\) 
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-69188908404875\)\(\medspace = -\,5^{3}\cdot 7^{5}\cdot 31\cdot 1062367\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $11.60$ | 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: | $5, 7, 31, 1062367$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $1$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1163} a^{12} - \frac{16}{1163} a^{11} + \frac{211}{1163} a^{10} - \frac{420}{1163} a^{9} - \frac{350}{1163} a^{8} - \frac{100}{1163} a^{7} + \frac{120}{1163} a^{6} - \frac{387}{1163} a^{5} + \frac{395}{1163} a^{4} - \frac{499}{1163} a^{3} - \frac{496}{1163} a^{2} - \frac{523}{1163} a - \frac{179}{1163}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$)
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 49.3240960032 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_{13}$ (as 13T9):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A non-solvable group of order 6227020800 |
| The 101 conjugacy class representatives for $S_{13}$ are not computed |
| Character table for $S_{13}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 26 sibling: | 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 | ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.13.0.1}{13} }$ | R | R | ${\href{/LocalNumberField/11.13.0.1}{13} }$ | ${\href{/LocalNumberField/13.13.0.1}{13} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.13.0.1}{13} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }$ | R | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.13.0.1}{13} }$ | ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.9.0.1 | $x^{9} + x^{2} - 2 x + 2$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| $7$ | 7.6.5.1 | $x^{6} - 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 31.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 1062367 | Data not computed | ||||||