Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157)
gp: K = bnfinit(y^16 - 7*y^15 + 49*y^14 - 292*y^13 + 1551*y^12 - 8273*y^11 + 27753*y^10 - 55551*y^9 + 67506*y^8 + 264153*y^7 + 606613*y^6 + 1111905*y^5 - 5899565*y^4 - 9805130*y^3 + 601264*y^2 + 18342111*y - 8394157, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157)
\( x^{16} - 7 x^{15} + 49 x^{14} - 292 x^{13} + 1551 x^{12} - 8273 x^{11} + 27753 x^{10} - 55551 x^{9} + \cdots - 8394157 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(698011763642789028435211838001152\)
\(\medspace = 2^{12}\cdot 13^{2}\cdot 17^{15}\cdot 137^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(112.91\) | 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: | $2^{7/4}13^{1/2}17^{15/16}137^{1/2}\approx 2021.5305826893875$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\), \(137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{58\!\cdots\!99}a^{15}+\frac{24\!\cdots\!80}{58\!\cdots\!99}a^{14}-\frac{10\!\cdots\!07}{58\!\cdots\!99}a^{13}+\frac{25\!\cdots\!23}{58\!\cdots\!99}a^{12}+\frac{20\!\cdots\!61}{58\!\cdots\!99}a^{11}-\frac{83\!\cdots\!59}{58\!\cdots\!99}a^{10}-\frac{22\!\cdots\!47}{58\!\cdots\!99}a^{9}+\frac{38\!\cdots\!39}{58\!\cdots\!99}a^{8}+\frac{10\!\cdots\!25}{58\!\cdots\!99}a^{7}+\frac{99\!\cdots\!76}{23\!\cdots\!49}a^{6}+\frac{14\!\cdots\!15}{58\!\cdots\!99}a^{5}-\frac{53\!\cdots\!30}{58\!\cdots\!99}a^{4}+\frac{12\!\cdots\!66}{58\!\cdots\!99}a^{3}+\frac{12\!\cdots\!36}{58\!\cdots\!99}a^{2}-\frac{28\!\cdots\!41}{58\!\cdots\!99}a-\frac{70\!\cdots\!71}{58\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| 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: |
$\frac{70\!\cdots\!93}{17\!\cdots\!09}a^{15}-\frac{57\!\cdots\!00}{17\!\cdots\!09}a^{14}+\frac{40\!\cdots\!71}{17\!\cdots\!09}a^{13}-\frac{24\!\cdots\!91}{17\!\cdots\!09}a^{12}+\frac{13\!\cdots\!35}{17\!\cdots\!09}a^{11}-\frac{68\!\cdots\!85}{17\!\cdots\!09}a^{10}+\frac{24\!\cdots\!50}{17\!\cdots\!09}a^{9}-\frac{53\!\cdots\!03}{17\!\cdots\!09}a^{8}+\frac{49\!\cdots\!17}{17\!\cdots\!09}a^{7}+\frac{31\!\cdots\!61}{17\!\cdots\!09}a^{6}-\frac{30\!\cdots\!74}{17\!\cdots\!09}a^{5}+\frac{12\!\cdots\!28}{17\!\cdots\!09}a^{4}-\frac{44\!\cdots\!26}{17\!\cdots\!09}a^{3}-\frac{43\!\cdots\!45}{17\!\cdots\!09}a^{2}+\frac{10\!\cdots\!47}{17\!\cdots\!09}a+\frac{44\!\cdots\!72}{17\!\cdots\!09}$, $\frac{95\!\cdots\!31}{17\!\cdots\!09}a^{15}-\frac{72\!\cdots\!89}{17\!\cdots\!09}a^{14}+\frac{49\!\cdots\!70}{17\!\cdots\!09}a^{13}-\frac{29\!\cdots\!62}{17\!\cdots\!09}a^{12}+\frac{15\!\cdots\!89}{17\!\cdots\!09}a^{11}-\frac{82\!\cdots\!74}{17\!\cdots\!09}a^{10}+\frac{28\!\cdots\!81}{17\!\cdots\!09}a^{9}-\frac{53\!\cdots\!50}{17\!\cdots\!09}a^{8}+\frac{30\!\cdots\!75}{17\!\cdots\!09}a^{7}+\frac{41\!\cdots\!22}{17\!\cdots\!09}a^{6}-\frac{41\!\cdots\!25}{17\!\cdots\!09}a^{5}+\frac{12\!\cdots\!94}{17\!\cdots\!09}a^{4}-\frac{69\!\cdots\!08}{17\!\cdots\!09}a^{3}-\frac{57\!\cdots\!11}{17\!\cdots\!09}a^{2}+\frac{15\!\cdots\!03}{17\!\cdots\!09}a-\frac{10\!\cdots\!33}{17\!\cdots\!09}$, $\frac{24\!\cdots\!38}{17\!\cdots\!09}a^{15}-\frac{14\!\cdots\!89}{17\!\cdots\!09}a^{14}+\frac{90\!\cdots\!99}{17\!\cdots\!09}a^{13}-\frac{50\!\cdots\!71}{17\!\cdots\!09}a^{12}+\frac{25\!\cdots\!54}{17\!\cdots\!09}a^{11}-\frac{13\!\cdots\!89}{17\!\cdots\!09}a^{10}+\frac{32\!\cdots\!31}{17\!\cdots\!09}a^{9}+\frac{75\!\cdots\!53}{17\!\cdots\!09}a^{8}-\frac{19\!\cdots\!42}{17\!\cdots\!09}a^{7}+\frac{10\!\cdots\!61}{17\!\cdots\!09}a^{6}+\frac{26\!\cdots\!49}{17\!\cdots\!09}a^{5}+\frac{75\!\cdots\!66}{17\!\cdots\!09}a^{4}-\frac{24\!\cdots\!82}{17\!\cdots\!09}a^{3}-\frac{14\!\cdots\!66}{17\!\cdots\!09}a^{2}+\frac{49\!\cdots\!56}{17\!\cdots\!09}a+\frac{12\!\cdots\!04}{17\!\cdots\!09}$, $\frac{88\!\cdots\!16}{23\!\cdots\!49}a^{15}-\frac{55\!\cdots\!86}{23\!\cdots\!49}a^{14}+\frac{39\!\cdots\!04}{23\!\cdots\!49}a^{13}-\frac{23\!\cdots\!50}{23\!\cdots\!49}a^{12}+\frac{12\!\cdots\!14}{23\!\cdots\!49}a^{11}-\frac{64\!\cdots\!88}{23\!\cdots\!49}a^{10}+\frac{19\!\cdots\!67}{23\!\cdots\!49}a^{9}-\frac{34\!\cdots\!59}{23\!\cdots\!49}a^{8}+\frac{33\!\cdots\!05}{23\!\cdots\!49}a^{7}+\frac{26\!\cdots\!04}{23\!\cdots\!49}a^{6}+\frac{70\!\cdots\!56}{23\!\cdots\!49}a^{5}+\frac{14\!\cdots\!83}{23\!\cdots\!49}a^{4}-\frac{43\!\cdots\!61}{23\!\cdots\!49}a^{3}-\frac{11\!\cdots\!68}{23\!\cdots\!49}a^{2}-\frac{77\!\cdots\!85}{23\!\cdots\!49}a+\frac{10\!\cdots\!00}{23\!\cdots\!49}$, $\frac{12\!\cdots\!71}{23\!\cdots\!49}a^{15}-\frac{83\!\cdots\!44}{23\!\cdots\!49}a^{14}+\frac{58\!\cdots\!75}{23\!\cdots\!49}a^{13}-\frac{34\!\cdots\!65}{23\!\cdots\!49}a^{12}+\frac{18\!\cdots\!07}{23\!\cdots\!49}a^{11}-\frac{96\!\cdots\!01}{23\!\cdots\!49}a^{10}+\frac{31\!\cdots\!99}{23\!\cdots\!49}a^{9}-\frac{56\!\cdots\!42}{23\!\cdots\!49}a^{8}+\frac{48\!\cdots\!92}{23\!\cdots\!49}a^{7}+\frac{42\!\cdots\!09}{23\!\cdots\!49}a^{6}+\frac{65\!\cdots\!92}{23\!\cdots\!49}a^{5}+\frac{19\!\cdots\!59}{23\!\cdots\!49}a^{4}-\frac{68\!\cdots\!23}{23\!\cdots\!49}a^{3}-\frac{13\!\cdots\!55}{23\!\cdots\!49}a^{2}-\frac{20\!\cdots\!52}{23\!\cdots\!49}a+\frac{67\!\cdots\!68}{23\!\cdots\!49}$, $\frac{10\!\cdots\!72}{23\!\cdots\!49}a^{15}-\frac{59\!\cdots\!86}{23\!\cdots\!49}a^{14}+\frac{40\!\cdots\!49}{23\!\cdots\!49}a^{13}-\frac{23\!\cdots\!49}{23\!\cdots\!49}a^{12}+\frac{11\!\cdots\!58}{23\!\cdots\!49}a^{11}-\frac{63\!\cdots\!43}{23\!\cdots\!49}a^{10}+\frac{17\!\cdots\!05}{23\!\cdots\!49}a^{9}-\frac{17\!\cdots\!22}{23\!\cdots\!49}a^{8}-\frac{61\!\cdots\!78}{23\!\cdots\!49}a^{7}+\frac{30\!\cdots\!11}{23\!\cdots\!49}a^{6}+\frac{12\!\cdots\!19}{23\!\cdots\!49}a^{5}+\frac{13\!\cdots\!24}{23\!\cdots\!49}a^{4}-\frac{62\!\cdots\!16}{23\!\cdots\!49}a^{3}-\frac{13\!\cdots\!13}{23\!\cdots\!49}a^{2}-\frac{70\!\cdots\!24}{23\!\cdots\!49}a+\frac{89\!\cdots\!18}{23\!\cdots\!49}$, $\frac{14\!\cdots\!72}{23\!\cdots\!49}a^{15}-\frac{89\!\cdots\!11}{23\!\cdots\!49}a^{14}+\frac{63\!\cdots\!09}{23\!\cdots\!49}a^{13}-\frac{37\!\cdots\!30}{23\!\cdots\!49}a^{12}+\frac{19\!\cdots\!29}{23\!\cdots\!49}a^{11}-\frac{10\!\cdots\!80}{23\!\cdots\!49}a^{10}+\frac{32\!\cdots\!62}{23\!\cdots\!49}a^{9}-\frac{57\!\cdots\!11}{23\!\cdots\!49}a^{8}+\frac{58\!\cdots\!43}{23\!\cdots\!49}a^{7}+\frac{41\!\cdots\!36}{23\!\cdots\!49}a^{6}+\frac{11\!\cdots\!79}{23\!\cdots\!49}a^{5}+\frac{24\!\cdots\!32}{23\!\cdots\!49}a^{4}-\frac{66\!\cdots\!96}{23\!\cdots\!49}a^{3}-\frac{19\!\cdots\!37}{23\!\cdots\!49}a^{2}-\frac{14\!\cdots\!58}{23\!\cdots\!49}a+\frac{15\!\cdots\!30}{23\!\cdots\!49}$, $\frac{31\!\cdots\!66}{58\!\cdots\!99}a^{15}-\frac{28\!\cdots\!72}{58\!\cdots\!99}a^{14}+\frac{20\!\cdots\!88}{58\!\cdots\!99}a^{13}-\frac{13\!\cdots\!06}{58\!\cdots\!99}a^{12}+\frac{74\!\cdots\!35}{58\!\cdots\!99}a^{11}-\frac{40\!\cdots\!95}{58\!\cdots\!99}a^{10}+\frac{16\!\cdots\!86}{58\!\cdots\!99}a^{9}-\frac{49\!\cdots\!70}{58\!\cdots\!99}a^{8}+\frac{11\!\cdots\!58}{58\!\cdots\!99}a^{7}-\frac{59\!\cdots\!54}{23\!\cdots\!49}a^{6}+\frac{48\!\cdots\!10}{58\!\cdots\!99}a^{5}-\frac{59\!\cdots\!54}{58\!\cdots\!99}a^{4}-\frac{67\!\cdots\!47}{58\!\cdots\!99}a^{3}-\frac{17\!\cdots\!80}{58\!\cdots\!99}a^{2}+\frac{36\!\cdots\!75}{58\!\cdots\!99}a-\frac{13\!\cdots\!54}{58\!\cdots\!99}$, $\frac{96\!\cdots\!43}{58\!\cdots\!99}a^{15}-\frac{46\!\cdots\!02}{58\!\cdots\!99}a^{14}+\frac{37\!\cdots\!03}{58\!\cdots\!99}a^{13}-\frac{19\!\cdots\!47}{58\!\cdots\!99}a^{12}+\frac{10\!\cdots\!96}{58\!\cdots\!99}a^{11}-\frac{56\!\cdots\!02}{58\!\cdots\!99}a^{10}+\frac{14\!\cdots\!30}{58\!\cdots\!99}a^{9}-\frac{21\!\cdots\!10}{58\!\cdots\!99}a^{8}+\frac{16\!\cdots\!48}{58\!\cdots\!99}a^{7}+\frac{11\!\cdots\!38}{23\!\cdots\!49}a^{6}+\frac{12\!\cdots\!33}{58\!\cdots\!99}a^{5}+\frac{38\!\cdots\!29}{58\!\cdots\!99}a^{4}+\frac{28\!\cdots\!86}{58\!\cdots\!99}a^{3}-\frac{30\!\cdots\!51}{58\!\cdots\!99}a^{2}-\frac{63\!\cdots\!31}{58\!\cdots\!99}a+\frac{36\!\cdots\!80}{58\!\cdots\!99}$
| 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: | \( 502651640.12 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 502651640.12 \cdot 32}{2\cdot\sqrt{698011763642789028435211838001152}}\cr\approx \mathstrut & 0.29967803233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 49*x^14 - 292*x^13 + 1551*x^12 - 8273*x^11 + 27753*x^10 - 55551*x^9 + 67506*x^8 + 264153*x^7 + 606613*x^6 + 1111905*x^5 - 5899565*x^4 - 9805130*x^3 + 601264*x^2 + 18342111*x - 8394157);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^7.C_8$ (as 16T1155):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.123226344856592.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | $16$ | $16$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.19 | $x^{8} - 2 x^{7} + 2 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ | $4$ | $2$ | $12$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(17\)
| 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(137\)
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.1.1 | $x^{2} + 137$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.2.1.1 | $x^{2} + 137$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.4.2.1 | $x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |