Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 - 25*y^14 + 92*y^13 + 98*y^12 - 583*y^11 + 49*y^10 + 1383*y^9 - 647*y^8 - 1357*y^7 + 807*y^6 + 520*y^5 - 251*y^4 - 114*y^3 + 19*y^2 + 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1)
\( x^{16} - 3 x^{15} - 25 x^{14} + 92 x^{13} + 98 x^{12} - 583 x^{11} + 49 x^{10} + 1383 x^{9} - 647 x^{8} + \cdots + 1 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(81753664774171848863873\)
\(\medspace = 13^{4}\cdot 17^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.04\) | 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: | $13^{1/2}17^{15/16}\approx 51.34729148263172$ | ||
| Ramified primes: |
\(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{103}a^{14}-\frac{25}{103}a^{13}+\frac{9}{103}a^{12}+\frac{22}{103}a^{11}+\frac{17}{103}a^{10}+\frac{51}{103}a^{9}+\frac{43}{103}a^{8}-\frac{26}{103}a^{7}-\frac{15}{103}a^{6}+\frac{29}{103}a^{5}-\frac{22}{103}a^{4}+\frac{48}{103}a^{3}-\frac{49}{103}a^{2}-\frac{11}{103}a+\frac{1}{103}$, $\frac{1}{2549095397}a^{15}-\frac{7272992}{2549095397}a^{14}+\frac{1027924379}{2549095397}a^{13}+\frac{1194144945}{2549095397}a^{12}+\frac{78928993}{2549095397}a^{11}+\frac{1239857956}{2549095397}a^{10}+\frac{396383097}{2549095397}a^{9}+\frac{155290479}{2549095397}a^{8}+\frac{341311969}{2549095397}a^{7}+\frac{1017202377}{2549095397}a^{6}-\frac{769072344}{2549095397}a^{5}+\frac{1223619063}{2549095397}a^{4}+\frac{973861671}{2549095397}a^{3}-\frac{1270128219}{2549095397}a^{2}+\frac{1007307724}{2549095397}a+\frac{534884425}{2549095397}$
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
Trivial group, which has order $1$ (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: | $15$ | 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: |
$\frac{6579520002}{2549095397}a^{15}-\frac{19984358257}{2549095397}a^{14}-\frac{162613400906}{2549095397}a^{13}+\frac{608801141020}{2549095397}a^{12}+\frac{592460444358}{2549095397}a^{11}-\frac{3775854314842}{2549095397}a^{10}+\frac{620220423320}{2549095397}a^{9}+\frac{8555228694011}{2549095397}a^{8}-\frac{4819970670709}{2549095397}a^{7}-\frac{7522028118263}{2549095397}a^{6}+\frac{5580411180913}{2549095397}a^{5}+\frac{19827201556}{24748499}a^{4}-\frac{1530341504332}{2549095397}a^{3}-\frac{298827581851}{2549095397}a^{2}+\frac{107365874562}{2549095397}a+\frac{18955080480}{2549095397}$, $\frac{9285927783}{2549095397}a^{15}-\frac{29798597636}{2549095397}a^{14}-\frac{225300057243}{2549095397}a^{13}+\frac{900085561514}{2549095397}a^{12}+\frac{705287388131}{2549095397}a^{11}-\frac{5518938094846}{2549095397}a^{10}+\frac{1705168231163}{2549095397}a^{9}+\frac{12214270857430}{2549095397}a^{8}-\frac{8760422975358}{2549095397}a^{7}-\frac{10119544229722}{2549095397}a^{6}+\frac{9753788945887}{2549095397}a^{5}+\frac{2165724089857}{2549095397}a^{4}-\frac{2807514528404}{2549095397}a^{3}-\frac{273001878104}{2549095397}a^{2}+\frac{250000721849}{2549095397}a+\frac{33067922121}{2549095397}$, $\frac{2193093357}{2549095397}a^{15}-\frac{7255557041}{2549095397}a^{14}-\frac{52316957658}{2549095397}a^{13}+\frac{217351233839}{2549095397}a^{12}+\frac{140660833855}{2549095397}a^{11}-\frac{1304348653824}{2549095397}a^{10}+\frac{550423645503}{2549095397}a^{9}+\frac{2754172422128}{2549095397}a^{8}-\frac{2342346938703}{2549095397}a^{7}-\frac{2014689649504}{2549095397}a^{6}+\frac{2413755314349}{2549095397}a^{5}+\frac{213291662191}{2549095397}a^{4}-\frac{579209085174}{2549095397}a^{3}-\frac{49166277492}{2549095397}a^{2}+\frac{46047096886}{2549095397}a+\frac{9112013295}{2549095397}$, $\frac{212207193}{2549095397}a^{15}+\frac{1630328359}{2549095397}a^{14}-\frac{10825581471}{2549095397}a^{13}-\frac{39276021859}{2549095397}a^{12}+\frac{193813782758}{2549095397}a^{11}+\frac{170288736204}{2549095397}a^{10}-\frac{1071808825073}{2549095397}a^{9}-\frac{73463030416}{2549095397}a^{8}+\frac{2332710209627}{2549095397}a^{7}-\frac{548758383942}{2549095397}a^{6}-\frac{2012781154804}{2549095397}a^{5}+\frac{655934581564}{2549095397}a^{4}+\frac{526374686893}{2549095397}a^{3}-\frac{77990475016}{2549095397}a^{2}-\frac{55799229067}{2549095397}a-\frac{4004668128}{2549095397}$, $\frac{4632279518}{2549095397}a^{15}-\frac{16016527879}{2549095397}a^{14}-\frac{108491044345}{2549095397}a^{13}+\frac{476287639073}{2549095397}a^{12}+\frac{235528425842}{2549095397}a^{11}-\frac{2821280489539}{2549095397}a^{10}+\frac{1543314061208}{2549095397}a^{9}+\frac{5778009041656}{2549095397}a^{8}-\frac{5808423710176}{2549095397}a^{7}-\frac{3801923554863}{2549095397}a^{6}+\frac{5860549636337}{2549095397}a^{5}-\frac{119160509457}{2549095397}a^{4}-\frac{1428161119737}{2549095397}a^{3}+\frac{80066223510}{2549095397}a^{2}+\frac{115706150219}{2549095397}a+\frac{10792473299}{2549095397}$, $\frac{9774492258}{2549095397}a^{15}-\frac{32082068659}{2549095397}a^{14}-\frac{235681753197}{2549095397}a^{13}+\frac{966563263756}{2549095397}a^{12}+\frac{695188191358}{2549095397}a^{11}-\frac{5920415553374}{2549095397}a^{10}+\frac{2089918127681}{2549095397}a^{9}+\frac{13096783353476}{2549095397}a^{8}-\frac{9889328169310}{2549095397}a^{7}-\frac{10890540681321}{2549095397}a^{6}+\frac{10847590061775}{2549095397}a^{5}+\frac{2451906206023}{2549095397}a^{4}-\frac{3104082129029}{2549095397}a^{3}-\frac{399609553753}{2549095397}a^{2}+\frac{277748862214}{2549095397}a+\frac{46914758521}{2549095397}$, $\frac{2788086269}{2549095397}a^{15}-\frac{7445713925}{2549095397}a^{14}-\frac{71904199454}{2549095397}a^{13}+\frac{232739682141}{2549095397}a^{12}+\frac{341831297202}{2549095397}a^{11}-\frac{1506892575870}{2549095397}a^{10}-\frac{279739877259}{2549095397}a^{9}+\frac{3692271746876}{2549095397}a^{8}-\frac{912658651706}{2549095397}a^{7}-\frac{3771464738253}{2549095397}a^{6}+\frac{1576469701481}{2549095397}a^{5}+\frac{1404998180817}{2549095397}a^{4}-\frac{614253197728}{2549095397}a^{3}-\frac{138287515336}{2549095397}a^{2}+\frac{49235359938}{2549095397}a+\frac{4625531467}{2549095397}$, $\frac{2119689325}{2549095397}a^{15}-\frac{7315943605}{2549095397}a^{14}-\frac{50117923417}{2549095397}a^{13}+\frac{218434966922}{2549095397}a^{12}+\frac{120661530545}{2549095397}a^{11}-\frac{1316332364826}{2549095397}a^{10}+\frac{628433531738}{2549095397}a^{9}+\frac{2811338862030}{2549095397}a^{8}-\frac{2484079751063}{2549095397}a^{7}-\frac{2122300065311}{2549095397}a^{6}+\frac{2527945858817}{2549095397}a^{5}+\frac{265458960719}{2549095397}a^{4}-\frac{608146088562}{2549095397}a^{3}-\frac{27692839377}{2549095397}a^{2}+\frac{42711696796}{2549095397}a+\frac{7181374915}{2549095397}$, $\frac{5364404540}{2549095397}a^{15}-\frac{18548837136}{2549095397}a^{14}-\frac{127513696369}{2549095397}a^{13}+\frac{556400598018}{2549095397}a^{12}+\frac{320714100815}{2549095397}a^{11}-\frac{3416637404200}{2549095397}a^{10}+\frac{1565000564280}{2549095397}a^{9}+\frac{7612558334664}{2549095397}a^{8}-\frac{63624518488}{24748499}a^{7}-\frac{6449139468964}{2549095397}a^{6}+\frac{7301425189679}{2549095397}a^{5}+\frac{1563072331981}{2549095397}a^{4}-\frac{2378360186842}{2549095397}a^{3}-\frac{271987707008}{2549095397}a^{2}+\frac{254255124097}{2549095397}a+\frac{43643520444}{2549095397}$, $\frac{15627484115}{2549095397}a^{15}-\frac{50878009377}{2549095397}a^{14}-\frac{376688235538}{2549095397}a^{13}+\frac{1531482246202}{2549095397}a^{12}+\frac{1114466408964}{2549095397}a^{11}-\frac{9316257183367}{2549095397}a^{10}+\frac{3268400205165}{2549095397}a^{9}+\frac{20279780017848}{2549095397}a^{8}-\frac{15406881216076}{2549095397}a^{7}-\frac{16131532739578}{2549095397}a^{6}+\frac{16506698881424}{2549095397}a^{5}+\frac{2904965386489}{2549095397}a^{4}-\frac{4322949278508}{2549095397}a^{3}-\frac{415765758045}{2549095397}a^{2}+\frac{354880078272}{2549095397}a+\frac{53364759957}{2549095397}$, $\frac{6102039738}{2549095397}a^{15}-\frac{21970891758}{2549095397}a^{14}-\frac{142372527705}{2549095397}a^{13}+\frac{653492823239}{2549095397}a^{12}+\frac{285170544728}{2549095397}a^{11}-\frac{3939080628550}{2549095397}a^{10}+\frac{2228154248463}{2549095397}a^{9}+\frac{8425442491198}{2549095397}a^{8}-\frac{8243154384083}{2549095397}a^{7}-\frac{6382311476946}{2549095397}a^{6}+\frac{8514238838137}{2549095397}a^{5}+\frac{817288896577}{2549095397}a^{4}-\frac{2312282561156}{2549095397}a^{3}-\frac{92609536624}{2549095397}a^{2}+\frac{192994767391}{2549095397}a+\frac{23099824893}{2549095397}$, $\frac{6389195952}{2549095397}a^{15}-\frac{21278010418}{2549095397}a^{14}-\frac{153206715339}{2549095397}a^{13}+\frac{639684561913}{2549095397}a^{12}+\frac{427914354091}{2549095397}a^{11}-\frac{3905925648498}{2549095397}a^{10}+\frac{1538907108217}{2549095397}a^{9}+\frac{8576164375152}{2549095397}a^{8}-\frac{6887427358770}{2549095397}a^{7}-\frac{6967450491423}{2549095397}a^{6}+\frac{7504875917662}{2549095397}a^{5}+\frac{1375687966835}{2549095397}a^{4}-\frac{2159028476679}{2549095397}a^{3}-\frac{187199785079}{2549095397}a^{2}+\frac{194842654543}{2549095397}a+\frac{24722422363}{2549095397}$, $\frac{2175855090}{2549095397}a^{15}-\frac{7913344736}{2549095397}a^{14}-\frac{49932251714}{2549095397}a^{13}+\frac{233608089680}{2549095397}a^{12}+\frac{78829931689}{2549095397}a^{11}-\frac{1368517518514}{2549095397}a^{10}+\frac{920794201297}{2549095397}a^{9}+\frac{2726531821651}{2549095397}a^{8}-\frac{3156076062761}{2549095397}a^{7}-\frac{1607901331086}{2549095397}a^{6}+\frac{3073085730575}{2549095397}a^{5}-\frac{283181847596}{2549095397}a^{4}-\frac{683738905150}{2549095397}a^{3}+\frac{79615584193}{2549095397}a^{2}+\frac{37471786439}{2549095397}a+\frac{2012690844}{2549095397}$, $\frac{8802111240}{2549095397}a^{15}-\frac{27605924457}{2549095397}a^{14}-\frac{215880233535}{2549095397}a^{13}+\frac{837843373290}{2549095397}a^{12}+\frac{738277149267}{2549095397}a^{11}-\frac{5190628040004}{2549095397}a^{10}+\frac{1174662498454}{2549095397}a^{9}+\frac{11747598554064}{2549095397}a^{8}-\frac{7256110567197}{2549095397}a^{7}-\frac{10314919535211}{2549095397}a^{6}+\frac{8232393930738}{2549095397}a^{5}+\frac{2810834233755}{2549095397}a^{4}-\frac{2315579073597}{2549095397}a^{3}-\frac{446665704063}{2549095397}a^{2}+\frac{193126831167}{2549095397}a+\frac{38558464157}{2549095397}$, $\frac{653178735}{2549095397}a^{15}-\frac{2661931622}{2549095397}a^{14}-\frac{14117184585}{2549095397}a^{13}+\frac{77471107711}{2549095397}a^{12}-\frac{3082414895}{2549095397}a^{11}-\frac{443947281150}{2549095397}a^{10}+\frac{449018046707}{2549095397}a^{9}+\frac{843330009899}{2549095397}a^{8}-\frac{1368569890022}{2549095397}a^{7}-\frac{412493022618}{2549095397}a^{6}+\frac{1350238034733}{2549095397}a^{5}-\frac{185642713034}{2549095397}a^{4}-\frac{366202360443}{2549095397}a^{3}+\frac{50884582949}{2549095397}a^{2}+\frac{30416086812}{2549095397}a-\frac{153864018}{2549095397}$
| 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: | \( 1330090.29679 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 1330090.29679 \cdot 1}{2\cdot\sqrt{81753664774171848863873}}\cr\approx \mathstrut & 0.152432455359 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1)
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 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1, 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 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1);
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 - 3*x^15 - 25*x^14 + 92*x^13 + 98*x^12 - 583*x^11 + 49*x^10 + 1383*x^9 - 647*x^8 - 1357*x^7 + 807*x^6 + 520*x^5 - 251*x^4 - 114*x^3 + 19*x^2 + 11*x + 1);
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
$\OD_{32}$ (as 16T22):
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 32 |
| The 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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
| Galois closure: | deg 32 |
| 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(17\)
| 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |