Normalized defining polynomial
\( x^{7} - x^{6} - 432x^{5} + 247x^{4} + 59784x^{3} - 15272x^{2} - 2647129x - 374699 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1055229678769825441\) \(\medspace = 1009^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(375.63\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $1009^{6/7}\approx 375.6331020554405$ | ||
Ramified primes: | \(1009\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $7$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1009\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1009}(1,·)$, $\chi_{1009}(935,·)$, $\chi_{1009}(105,·)$, $\chi_{1009}(394,·)$, $\chi_{1009}(859,·)$, $\chi_{1009}(302,·)$, $\chi_{1009}(431,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23816897663}a^{6}+\frac{3015398990}{23816897663}a^{5}+\frac{812623947}{1832069051}a^{4}-\frac{645249724}{1832069051}a^{3}-\frac{2730071282}{23816897663}a^{2}-\frac{5566198370}{23816897663}a+\frac{3861739}{44684611}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{83649492}{23816897663}a^{6}-\frac{262121511}{23816897663}a^{5}-\frac{1921890103}{1832069051}a^{4}+\frac{5526471218}{1832069051}a^{3}+\frac{2272124589957}{23816897663}a^{2}-\frac{4297538708067}{23816897663}a-\frac{119004781542}{44684611}$, $\frac{1691935301}{23816897663}a^{6}+\frac{2945471176}{23816897663}a^{5}-\frac{38922658853}{1832069051}a^{4}-\frac{125263226443}{1832069051}a^{3}+\frac{34970021181291}{23816897663}a^{2}+\frac{156674692186486}{23816897663}a+\frac{40334927299}{44684611}$, $\frac{33350160}{23816897663}a^{6}-\frac{1207754866}{23816897663}a^{5}-\frac{1106372171}{1832069051}a^{4}+\frac{30391470612}{1832069051}a^{3}+\frac{2066150502796}{23816897663}a^{2}-\frac{31166419120793}{23816897663}a-\frac{158868611177}{44684611}$, $\frac{60767362}{1832069051}a^{6}+\frac{581769194}{1832069051}a^{5}-\frac{19515356644}{1832069051}a^{4}-\frac{189645784658}{1832069051}a^{3}+\frac{1456306512146}{1832069051}a^{2}+\frac{14203690232267}{1832069051}a+\frac{48373297878}{44684611}$, $\frac{37062863964}{23816897663}a^{6}-\frac{384206980181}{23816897663}a^{5}-\frac{954786740399}{1832069051}a^{4}+\frac{9646650484403}{1832069051}a^{3}+\frac{10\!\cdots\!08}{23816897663}a^{2}-\frac{10\!\cdots\!78}{23816897663}a-\frac{2781870897361}{44684611}$, $\frac{101438362}{23816897663}a^{6}+\frac{4368022874}{23816897663}a^{5}-\frac{2542239961}{1832069051}a^{4}-\frac{110821947134}{1832069051}a^{3}+\frac{2401206084254}{23816897663}a^{2}+\frac{112229687571292}{23816897663}a-\frac{68067370144}{44684611}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 8038322.35166 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 8038322.35166 \cdot 1}{2\cdot\sqrt{1055229678769825441}}\cr\approx \mathstrut & 0.500808733101 \end{aligned}\]
Galois group
A cyclic group of order 7 |
The 7 conjugacy class representatives for $C_7$ |
Character table for $C_7$ |
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.1.0.1}{1} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.1.0.1}{1} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.1.0.1}{1} }^{7}$ | ${\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.1.0.1}{1} }^{7}$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(1009\) | Deg $7$ | $7$ | $1$ | $6$ |