Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1)
gp: K = bnfinit(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, -1, -5, 9, -3, -12, 19, -9, -10, 18, -11, -1, 6, -4, 1]);
\( x^{15} - 4 x^{14} + 6 x^{13} - x^{12} - 11 x^{11} + 18 x^{10} - 10 x^{9} - 9 x^{8} + 19 x^{7} - 12 x^{6} - 3 x^{5} + 9 x^{4} - 5 x^{3} - x^{2} + 3 x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-10496055636998343\)\(\medspace = -\,3^{5}\cdot 157\cdot 2377\cdot 115742009\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $11.70$ | 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: | $3, 157, 2377, 115742009$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
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: | $9$ | 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: | \( 99.2990074596 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A non-solvable group of order 1307674368000 |
| The 176 conjugacy class representatives for S15 are not computed |
| Character table for S15 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 30 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 | $15$ | R | ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | $15$ | $15$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | $15$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.5.0.1 | $x^{5} - x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 3.10.5.1 | $x^{10} - 18 x^{6} + 81 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| $157$ | 157.2.1.2 | $x^{2} + 785$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 157.13.0.1 | $x^{13} - x + 6$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| 2377 | Data not computed | ||||||
| 115742009 | Data not computed | ||||||