Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033)
gp: K = bnfinit(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13033, 11592, 9737, -5862, -2038, -3952, 2447, -176, 110, 12, -83, 20, 2, -2, 1]);
\(x^{14} - 2 x^{13} + 2 x^{12} + 20 x^{11} - 83 x^{10} + 12 x^{9} + 110 x^{8} - 176 x^{7} + 2447 x^{6} - 3952 x^{5} - 2038 x^{4} - 5862 x^{3} + 9737 x^{2} + 11592 x + 13033\) 
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-5796901408038404767744\)\(\medspace = -\,2^{14}\cdot 29^{12}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $35.85$ | 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: | $2, 29$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $7$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{12789774316960076723418958513} a^{13} + \frac{3444104340696083908158094412}{12789774316960076723418958513} a^{12} - \frac{4739613857555655773131208150}{12789774316960076723418958513} a^{11} - \frac{4291071350582826000407388714}{12789774316960076723418958513} a^{10} - \frac{2562070588677547455380550548}{12789774316960076723418958513} a^{9} - \frac{3402422768623145219542832622}{12789774316960076723418958513} a^{8} + \frac{3026263519484020038696007711}{12789774316960076723418958513} a^{7} + \frac{4118663202952402147778047183}{12789774316960076723418958513} a^{6} + \frac{3864131047626249020164117198}{12789774316960076723418958513} a^{5} - \frac{4174668285972796715749954853}{12789774316960076723418958513} a^{4} + \frac{3601759206183461787477556134}{12789774316960076723418958513} a^{3} - \frac{697608557525749742143957990}{12789774316960076723418958513} a^{2} - \frac{4261980420607665427919548078}{12789774316960076723418958513} a + \frac{5201595678199828828944356224}{12789774316960076723418958513}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
$C_{7}$, which has order $7$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( \frac{12317228248971}{163719440972021573} a^{13} - \frac{26878028333909}{163719440972021573} a^{12} + \frac{92349634788534}{163719440972021573} a^{11} + \frac{204932961825272}{163719440972021573} a^{10} - \frac{945241267907748}{163719440972021573} a^{9} + \frac{1863823155722795}{163719440972021573} a^{8} - \frac{1854211622966132}{163719440972021573} a^{7} - \frac{4444186303339562}{163719440972021573} a^{6} + \frac{33985587918655108}{163719440972021573} a^{5} - \frac{70680160761132613}{163719440972021573} a^{4} + \frac{120436235845816693}{163719440972021573} a^{3} - \frac{134346294211846567}{163719440972021573} a^{2} - \frac{37553476289553289}{163719440972021573} a - \frac{218804396285973327}{163719440972021573} \) (order $4$)
| 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: | \( 14379.655158793385 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_7\times D_7$ (as 14T8):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 98 |
| The 35 conjugacy class representatives for $C_7 \wr C_2$ |
| Character table for $C_7 \wr C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 14 siblings: | 14.0.9745585291264.1, Deg 14 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.14.0.1}{14} }$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/7.14.0.1}{14} }$ | ${\href{/LocalNumberField/11.14.0.1}{14} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }$ | ${\href{/LocalNumberField/23.14.0.1}{14} }$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.14.14.38 | $x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
| $29$ | 29.7.6.5 | $x^{7} + 58$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.6.6 | $x^{7} - 464$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.116.14t1.b.c | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.d | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.e | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.b | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.b.a | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.116.14t1.b.f | $1$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ | |
| 2.3364.7t2.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.d | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.f | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.14t8.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.3364.7t2.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| 2.3364.14t8.a.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.e | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.a | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.3364.14t8.b.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.116.14t8.a.b | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| 2.116.14t8.a.c | $2$ | $ 2^{2} \cdot 29 $ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ | |
| * | 2.3364.14t8.b.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.7t2.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 7.1.38068692544.1 | $D_{7}$ (as 7T2) | $1$ | $0$ | |
| * | 2.3364.14t8.b.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| * | 2.3364.14t8.b.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
| 2.3364.14t8.a.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 14.0.5796901408038404767744.4 | $C_7 \wr C_2$ (as 14T8) | $0$ | $0$ |
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 *.