Normalized defining polynomial
\( x^{7} - 609x^{5} - 2233x^{4} + 36743x^{3} + 62118x^{2} - 576520x + 3625 \)
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: | \(8233120419813614521\) \(\medspace = 7^{12}\cdot 29^{6}\) | 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: | \(503.76\) | 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: | $7^{12/7}29^{6/7}\approx 503.75978549322235$ | ||
Ramified primes: | \(7\), \(29\) | 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)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1421=7^{2}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(1051,·)$, $\chi_{1421}(1156,·)$, $\chi_{1421}(596,·)$, $\chi_{1421}(1387,·)$, $\chi_{1421}(1212,·)$, $\chi_{1421}(484,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{215}a^{5}+\frac{19}{215}a^{4}-\frac{17}{215}a^{3}-\frac{59}{215}a^{2}-\frac{39}{215}a-\frac{5}{43}$, $\frac{1}{24321428875}a^{6}+\frac{8416536}{24321428875}a^{5}+\frac{11875818}{223132375}a^{4}-\frac{10840722801}{24321428875}a^{3}-\frac{765479643}{24321428875}a^{2}-\frac{694599251}{4864285775}a-\frac{24660061}{194571431}$
Monogenic: | No | |
Index: | $25$ | |
Inessential primes: | $5$ |
Class group and class number
$C_{7}$, which has order $7$
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: | $\frac{372894817}{24321428875}a^{6}-\frac{7698310838}{24321428875}a^{5}-\frac{628602569}{223132375}a^{4}+\frac{589139102833}{24321428875}a^{3}+\frac{1604389212294}{24321428875}a^{2}-\frac{2102998468992}{4864285775}a+\frac{4353384499}{194571431}$, $\frac{46344533}{24321428875}a^{6}-\frac{424138687}{24321428875}a^{5}-\frac{223768631}{223132375}a^{4}+\frac{120227580567}{24321428875}a^{3}+\frac{629068789656}{24321428875}a^{2}-\frac{610566262958}{4864285775}a+\frac{4076830424}{194571431}$, $\frac{15999533}{4864285775}a^{6}+\frac{405159033}{4864285775}a^{5}+\frac{4341129}{44626475}a^{4}-\frac{24428139378}{4864285775}a^{3}-\frac{25415077259}{4864285775}a^{2}+\frac{73736509787}{972857155}a-\frac{3024837}{4524917}$, $\frac{1678454}{24321428875}a^{6}-\frac{22358656}{24321428875}a^{5}-\frac{4847528}{223132375}a^{4}+\frac{63599646}{24321428875}a^{3}+\frac{10505630553}{24321428875}a^{2}+\frac{887898536}{4864285775}a-\frac{226677}{194571431}$, $\frac{127621589}{24321428875}a^{6}-\frac{678500271}{24321428875}a^{5}-\frac{679876023}{223132375}a^{4}+\frac{109086533236}{24321428875}a^{3}+\frac{4097988073573}{24321428875}a^{2}-\frac{2765987906624}{4864285775}a+\frac{694854000}{194571431}$, $\frac{207960592}{24321428875}a^{6}+\frac{3581662}{24321428875}a^{5}-\frac{1162239794}{223132375}a^{4}-\frac{466212879092}{24321428875}a^{3}+\frac{7651675778294}{24321428875}a^{2}+\frac{2589965303443}{4864285775}a-\frac{959235859735}{194571431}$ | 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: | \( 5753883.56452 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 5753883.56452 \cdot 7}{2\cdot\sqrt{8233120419813614521}}\cr\approx \mathstrut & 0.898373327797 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.1.0.1}{1} }^{7}$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.7.0.1}{7} }$ | R | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.1.0.1}{1} }^{7}$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.7.12.3 | $x^{7} + 42 x^{6} + 105$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
\(29\) | 29.7.6.1 | $x^{7} + 232$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |