Normalized defining polynomial
\( x^{14} - 6x^{12} + 59x^{10} - 284x^{8} + 1027x^{6} - 1306x^{4} + 2445x^{2} + 511 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-9098007718612700671\) \(\medspace = -\,7^{7}\cdot 73^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.61\) | 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: | $7^{1/2}73^{1/2}\approx 22.60530911091463$ | ||
Ramified primes: | \(7\), \(73\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-511}) \) | ||
$\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{42}a^{10}+\frac{1}{21}a^{8}-\frac{2}{21}a^{6}+\frac{1}{21}a^{4}-\frac{1}{2}a^{3}+\frac{1}{42}a^{2}-\frac{1}{2}a$, $\frac{1}{42}a^{11}+\frac{1}{21}a^{9}+\frac{1}{14}a^{7}+\frac{3}{14}a^{5}-\frac{1}{2}a^{4}+\frac{4}{21}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{520460262}a^{12}+\frac{370117}{260230131}a^{10}-\frac{1182715}{86743377}a^{8}+\frac{23857241}{260230131}a^{6}-\frac{1}{2}a^{5}-\frac{477}{2753758}a^{4}-\frac{1}{2}a^{3}+\frac{115514509}{260230131}a^{2}+\frac{13139060}{37175733}$, $\frac{1}{520460262}a^{13}+\frac{370117}{260230131}a^{11}-\frac{1182715}{86743377}a^{9}-\frac{39028895}{520460262}a^{7}-\frac{1}{6}a^{6}-\frac{689155}{4130637}a^{5}-\frac{1}{6}a^{4}+\frac{144285641}{520460262}a^{3}+\frac{1}{3}a^{2}+\frac{13886209}{74351466}a-\frac{1}{6}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$
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)
| |
Fundamental units: | $\frac{7141}{37175733}a^{13}+\frac{8986}{37175733}a^{12}-\frac{24184}{37175733}a^{11}-\frac{116147}{74351466}a^{10}+\frac{228581}{24783822}a^{9}+\frac{417953}{24783822}a^{8}-\frac{3048607}{74351466}a^{7}-\frac{631723}{10621638}a^{6}+\frac{1116107}{8261274}a^{5}+\frac{1525807}{8261274}a^{4}-\frac{1150717}{37175733}a^{3}+\frac{338833}{74351466}a^{2}+\frac{1346647}{5310819}a+\frac{3781669}{5310819}$, $\frac{7141}{37175733}a^{13}-\frac{8986}{37175733}a^{12}-\frac{24184}{37175733}a^{11}+\frac{116147}{74351466}a^{10}+\frac{228581}{24783822}a^{9}-\frac{417953}{24783822}a^{8}-\frac{3048607}{74351466}a^{7}+\frac{631723}{10621638}a^{6}+\frac{1116107}{8261274}a^{5}-\frac{1525807}{8261274}a^{4}-\frac{1150717}{37175733}a^{3}-\frac{338833}{74351466}a^{2}+\frac{1346647}{5310819}a-\frac{3781669}{5310819}$, $\frac{138962}{260230131}a^{13}+\frac{25805}{260230131}a^{12}-\frac{856103}{260230131}a^{11}-\frac{784873}{520460262}a^{10}+\frac{2947526}{86743377}a^{9}+\frac{1134235}{173486754}a^{8}-\frac{39971086}{260230131}a^{7}-\frac{38097455}{520460262}a^{6}+\frac{366158}{590091}a^{5}+\frac{99179}{1180182}a^{4}-\frac{388710355}{520460262}a^{3}-\frac{56964521}{260230131}a^{2}+\frac{48953693}{37175733}a-\frac{3267142}{37175733}$, $\frac{43949}{86743377}a^{13}+\frac{101047}{520460262}a^{12}-\frac{352946}{86743377}a^{11}+\frac{425101}{260230131}a^{10}+\frac{1801013}{57828918}a^{9}-\frac{295205}{173486754}a^{8}-\frac{30608477}{173486754}a^{7}+\frac{47274151}{520460262}a^{6}+\frac{331390}{590091}a^{5}-\frac{594418}{1376879}a^{4}-\frac{53412077}{86743377}a^{3}+\frac{366876646}{260230131}a^{2}-\frac{12581633}{24783822}a+\frac{18035102}{37175733}$, $\frac{121853}{173486754}a^{13}-\frac{276671}{520460262}a^{12}-\frac{986567}{173486754}a^{11}-\frac{167917}{520460262}a^{10}+\frac{2484949}{57828918}a^{9}-\frac{1990738}{86743377}a^{8}-\frac{45122677}{173486754}a^{7}+\frac{866077}{520460262}a^{6}+\frac{2262263}{2753758}a^{5}-\frac{208317}{2753758}a^{4}-\frac{95496346}{86743377}a^{3}-\frac{3512203}{520460262}a^{2}+\frac{43390967}{24783822}a+\frac{7398323}{37175733}$, $\frac{51635}{57828918}a^{13}+\frac{107263}{74351466}a^{12}-\frac{89225}{28914459}a^{11}-\frac{40361}{5310819}a^{10}+\frac{2235967}{57828918}a^{9}+\frac{924863}{12391911}a^{8}-\frac{2440427}{19276306}a^{7}-\frac{25395623}{74351466}a^{6}+\frac{2189059}{8261274}a^{5}+\frac{3985060}{4130637}a^{4}+\frac{22065044}{28914459}a^{3}-\frac{37262089}{74351466}a^{2}-\frac{1110965}{4130637}a-\frac{2356069}{5310819}$ | 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: | \( 6924.33979828 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 6924.33979828 \cdot 2}{2\cdot\sqrt{9098007718612700671}}\cr\approx \mathstrut & 0.887491699524 \end{aligned}\]
Galois group
A solvable group of order 14 |
The 5 conjugacy class representatives for $D_{7}$ |
Character table for $D_{7}$ |
Intermediate fields
\(\Q(\sqrt{-511}) \), 7.1.133432831.1 x7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 7 sibling: | 7.1.133432831.1 |
Minimal sibling: | 7.1.133432831.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{7}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{7}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{7}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(73\) | 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.511.2t1.a.a | $1$ | $ 7 \cdot 73 $ | \(\Q(\sqrt{-511}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
*2 | 2.511.7t2.b.b | $2$ | $ 7 \cdot 73 $ | 14.0.9098007718612700671.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |
*2 | 2.511.7t2.b.c | $2$ | $ 7 \cdot 73 $ | 14.0.9098007718612700671.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |
*2 | 2.511.7t2.b.a | $2$ | $ 7 \cdot 73 $ | 14.0.9098007718612700671.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |