Normalized defining polynomial
\( x^{7} - 903x^{5} - 11739x^{4} - 4515x^{3} + 484008x^{2} + 1555568x - 609267 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(87495801462998035849\) \(\medspace = 7^{12}\cdot 43^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(706.08\) | 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: | $7^{12/7}43^{6/7}\approx 706.0822861068045$ | ||
Ramified primes: | \(7\), \(43\) | 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: | \(2107=7^{2}\cdot 43\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(274,·)$, $\chi_{2107}(1331,·)$, $\chi_{2107}(1268,·)$, $\chi_{2107}(1681,·)$, $\chi_{2107}(183,·)$, $\chi_{2107}(1884,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{1431}a^{5}+\frac{3}{53}a^{4}+\frac{109}{1431}a^{3}-\frac{61}{477}a^{2}-\frac{263}{1431}a-\frac{17}{477}$, $\frac{1}{926725617}a^{6}-\frac{119660}{926725617}a^{5}+\frac{40075561}{926725617}a^{4}+\frac{54518437}{926725617}a^{3}+\frac{403521403}{926725617}a^{2}-\frac{288745934}{926725617}a+\frac{19074514}{308908539}$
Monogenic: | No | |
Index: | $81$ | |
Inessential primes: | $3$ |
Class group and class number
$C_{7}$, which has order $7$ (assuming GRH)
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: | $\frac{18646276}{308908539}a^{6}-\frac{198852269}{308908539}a^{5}-\frac{14742058730}{308908539}a^{4}-\frac{61283165201}{308908539}a^{3}+\frac{587324327641}{308908539}a^{2}+\frac{2764181855245}{308908539}a-\frac{379918763906}{102969513}$, $\frac{39290423}{926725617}a^{6}-\frac{17383825}{926725617}a^{5}-\frac{35288363182}{926725617}a^{4}-\frac{446662035187}{926725617}a^{3}-\frac{140163941896}{926725617}a^{2}+\frac{17870914114379}{926725617}a+\frac{19834718494913}{308908539}$, $\frac{1080311395}{926725617}a^{6}-\frac{7039880918}{926725617}a^{5}-\frac{929648055848}{926725617}a^{4}-\frac{6623712043517}{926725617}a^{3}+\frac{38288338895554}{926725617}a^{2}+\frac{273416906570896}{926725617}a-\frac{33672296224232}{308908539}$, $\frac{115596635}{102969513}a^{6}-\frac{2138040746}{308908539}a^{5}-\frac{99991061746}{102969513}a^{4}-\frac{2221577733149}{308908539}a^{3}+\frac{4044669631375}{102969513}a^{2}+\frac{93091198862266}{308908539}a-\frac{11452458815396}{102969513}$, $\frac{1518358}{308908539}a^{6}-\frac{21297854}{308908539}a^{5}-\frac{1465045862}{308908539}a^{4}-\frac{9519704150}{308908539}a^{3}+\frac{59308845793}{308908539}a^{2}+\frac{417988405789}{308908539}a-\frac{51520354805}{102969513}$, $\frac{220995596}{926725617}a^{6}+\frac{1396947986}{926725617}a^{5}-\frac{190730533381}{926725617}a^{4}-\frac{3799612426921}{926725617}a^{3}-\frac{25007419839565}{926725617}a^{2}-\frac{51041423031310}{926725617}a+\frac{7104075158708}{308908539}$ (assuming GRH) | 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: | \( 80239711.6814 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 80239711.6814 \cdot 7}{2\cdot\sqrt{87495801462998035849}}\cr\approx \mathstrut & 3.84302992874 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.1.0.1}{1} }^{7}$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | R | ${\href{/padicField/47.1.0.1}{1} }^{7}$ | ${\href{/padicField/53.1.0.1}{1} }^{7}$ | ${\href{/padicField/59.7.0.1}{7} }$ |
In the table, R denotes a ramified prime. 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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
\(43\) | 43.7.6.5 | $x^{7} + 129$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.2107.7t1.a.a | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.2107.7t1.a.b | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.2107.7t1.a.c | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.2107.7t1.a.d | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.2107.7t1.a.e | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |
* | 1.2107.7t1.a.f | $1$ | $ 7^{2} \cdot 43 $ | 7.7.87495801462998035849.1 | $C_7$ (as 7T1) | $0$ | $1$ |