Normalized defining polynomial
\( x^{14} + 10x^{12} + 31x^{10} - 28x^{8} - 281x^{6} + 3950x^{4} + 29709x^{2} + 50519 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1238256615687209381111\) \(\medspace = -\,1031^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.11\) | 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: | $1031^{1/2}\approx 32.109188716004645$ | ||
Ramified primes: | \(1031\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1031}) \) | ||
$\card{ \Gal(K/\Q) }$: | $14$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{8}-\frac{1}{2}a^{6}+\frac{1}{26}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{6}{13}$, $\frac{1}{26}a^{9}-\frac{6}{13}a^{5}-\frac{1}{2}a^{2}+\frac{1}{26}a-\frac{1}{2}$, $\frac{1}{26}a^{10}-\frac{6}{13}a^{6}-\frac{1}{2}a^{3}+\frac{1}{26}a^{2}-\frac{1}{2}a$, $\frac{1}{26}a^{11}+\frac{1}{26}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{6}{13}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1274891722}a^{12}+\frac{169373}{98068594}a^{10}-\frac{1006321}{637445861}a^{8}+\frac{15471207}{49034297}a^{6}-\frac{1}{2}a^{5}+\frac{297789159}{1274891722}a^{4}-\frac{21971785}{98068594}a^{2}-\frac{1}{2}a+\frac{37021519}{637445861}$, $\frac{1}{8924242054}a^{13}+\frac{1970621}{343240079}a^{11}+\frac{145090249}{8924242054}a^{9}-\frac{23060415}{98068594}a^{7}-\frac{1}{2}a^{6}+\frac{1497337747}{4462121027}a^{5}-\frac{1}{2}a^{4}+\frac{128902975}{686480158}a^{3}-\frac{1}{2}a^{2}-\frac{208149966}{4462121027}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $13$ |
Class group and class number
$C_{5}$, which has order $5$
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: | $\frac{1620183}{8924242054}a^{13}+\frac{183858}{637445861}a^{12}+\frac{234951}{343240079}a^{11}+\frac{30570}{49034297}a^{10}-\frac{18568689}{8924242054}a^{9}-\frac{6021051}{1274891722}a^{8}-\frac{2394943}{98068594}a^{7}-\frac{1590894}{49034297}a^{6}+\frac{8349447}{4462121027}a^{5}+\frac{56748271}{637445861}a^{4}+\frac{715328891}{686480158}a^{3}+\frac{96408540}{49034297}a^{2}+\frac{26214557003}{8924242054}a+\frac{2660856532}{637445861}$, $\frac{44691}{343240079}a^{13}+\frac{1237035}{1274891722}a^{12}+\frac{858505}{686480158}a^{11}+\frac{549843}{98068594}a^{10}+\frac{776421}{343240079}a^{9}+\frac{7833705}{1274891722}a^{8}-\frac{224993}{49034297}a^{7}-\frac{5843999}{98068594}a^{6}-\frac{13940597}{686480158}a^{5}-\frac{35750199}{1274891722}a^{4}+\frac{435286213}{686480158}a^{3}+\frac{386918761}{98068594}a^{2}+\frac{1075492092}{343240079}a+\frac{7511394437}{637445861}$, $\frac{7009351}{8924242054}a^{13}+\frac{464701}{1274891722}a^{12}+\frac{108117}{26403083}a^{11}+\frac{106025}{49034297}a^{10}+\frac{24584373}{4462121027}a^{9}+\frac{1715468}{637445861}a^{8}-\frac{159509}{3771869}a^{7}-\frac{1170695}{98068594}a^{6}-\frac{17017031}{4462121027}a^{5}-\frac{27442570}{637445861}a^{4}+\frac{160186143}{52806166}a^{3}+\frac{71346792}{49034297}a^{2}+\frac{35025877974}{4462121027}a+\frac{2852616110}{637445861}$, $\frac{1012846}{637445861}a^{12}+\frac{391569}{49034297}a^{10}+\frac{6430049}{637445861}a^{8}-\frac{4806013}{49034297}a^{6}-\frac{11534404}{637445861}a^{4}+\frac{301843148}{49034297}a^{2}+\frac{10114310699}{637445861}$, $\frac{1620183}{8924242054}a^{13}-\frac{183858}{637445861}a^{12}+\frac{234951}{343240079}a^{11}-\frac{30570}{49034297}a^{10}-\frac{18568689}{8924242054}a^{9}+\frac{6021051}{1274891722}a^{8}-\frac{2394943}{98068594}a^{7}+\frac{1590894}{49034297}a^{6}+\frac{8349447}{4462121027}a^{5}-\frac{56748271}{637445861}a^{4}+\frac{715328891}{686480158}a^{3}-\frac{96408540}{49034297}a^{2}+\frac{26214557003}{8924242054}a-\frac{2660856532}{637445861}$, $\frac{169628}{637445861}a^{12}+\frac{77071}{49034297}a^{10}+\frac{1711373}{1274891722}a^{8}-\frac{2256929}{98068594}a^{6}-\frac{25178009}{1274891722}a^{4}+\frac{92447421}{98068594}a^{2}-\frac{1}{2}a+\frac{2138022461}{637445861}$ | 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: | \( 18383.6173323 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 18383.6173323 \cdot 5}{2\cdot\sqrt{1238256615687209381111}}\cr\approx \mathstrut & 0.504922507132 \end{aligned}\]
Galois group
A solvable group of order 14 |
The 5 conjugacy class representatives for $D_{7}$ |
Character table for $D_{7}$ |
Intermediate fields
\(\Q(\sqrt{-1031}) \), 7.1.1095912791.1 x7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 7 sibling: | 7.1.1095912791.1 |
Minimal sibling: | 7.1.1095912791.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{7}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.1.0.1}{1} }^{14}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(1031\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |