Normalized defining polynomial
\( x^{7} - x^{6} - 642x^{5} + 367x^{4} + 114514x^{3} - 287002x^{2} - 6382709x + 28440251 \)
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)
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Discriminant: | \(11345138370033741001\) \(\medspace = 1499^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(527.37\) | 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))
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Galois root discriminant: | $1499^{6/7}\approx 527.3703211123925$ | ||
Ramified primes: | \(1499\) | 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: | \(1499\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1499}(1,·)$, $\chi_{1499}(546,·)$, $\chi_{1499}(151,·)$, $\chi_{1499}(922,·)$, $\chi_{1499}(316,·)$, $\chi_{1499}(1314,·)$, $\chi_{1499}(1247,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{41}a^{5}+\frac{15}{41}a^{4}+\frac{1}{41}a^{3}-\frac{9}{41}a^{2}+\frac{14}{41}a-\frac{2}{41}$, $\frac{1}{80362278493}a^{6}+\frac{365349717}{80362278493}a^{5}+\frac{23938801534}{80362278493}a^{4}+\frac{465052181}{80362278493}a^{3}-\frac{1100971226}{80362278493}a^{2}-\frac{1501746453}{80362278493}a+\frac{1912473901}{11480325499}$
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)
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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)
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Fundamental units: | $\frac{1937759777}{11480325499}a^{6}+\frac{15396464628}{11480325499}a^{5}-\frac{1108223931567}{11480325499}a^{4}-\frac{9203316287300}{11480325499}a^{3}+\frac{140207855815646}{11480325499}a^{2}+\frac{696349257943077}{11480325499}a-\frac{61\!\cdots\!74}{11480325499}$, $\frac{206717087}{80362278493}a^{6}+\frac{919296904}{80362278493}a^{5}-\frac{130176052127}{80362278493}a^{4}-\frac{711532392903}{80362278493}a^{3}+\frac{19762044789633}{80362278493}a^{2}+\frac{62123282855412}{80362278493}a-\frac{136157694625418}{11480325499}$, $\frac{2657373377}{80362278493}a^{6}+\frac{24062227593}{80362278493}a^{5}-\frac{1669968653769}{80362278493}a^{4}-\frac{14726192521370}{80362278493}a^{3}+\frac{270228140757640}{80362278493}a^{2}+\frac{14\!\cdots\!95}{80362278493}a-\frac{23\!\cdots\!78}{11480325499}$, $\frac{2916990292}{80362278493}a^{6}+\frac{49799219703}{80362278493}a^{5}-\frac{680125906765}{80362278493}a^{4}-\frac{9961775279434}{80362278493}a^{3}+\frac{71319356152623}{80362278493}a^{2}+\frac{525106323176390}{80362278493}a-\frac{466418015195410}{11480325499}$, $\frac{721127227}{11480325499}a^{6}+\frac{6542625267}{11480325499}a^{5}-\frac{468578560076}{11480325499}a^{4}-\frac{3638763197359}{11480325499}a^{3}+\frac{75724980369705}{11480325499}a^{2}+\frac{305946522286170}{11480325499}a-\frac{39\!\cdots\!87}{11480325499}$, $\frac{2447391600}{80362278493}a^{6}+\frac{14352953357}{80362278493}a^{5}-\frac{1470523466757}{80362278493}a^{4}-\frac{9151760417236}{80362278493}a^{3}+\frac{216368411818369}{80362278493}a^{2}+\frac{759815775447773}{80362278493}a-\frac{14\!\cdots\!69}{11480325499}$ | 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: | \( 22758620.4726 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 22758620.4726 \cdot 1}{2\cdot\sqrt{11345138370033741001}}\cr\approx \mathstrut & 0.432435171000 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.1.0.1}{1} }^{7}$ | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.1.0.1}{1} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(1499\) | Deg $7$ | $7$ | $1$ | $6$ |