y2+xy+y=x3+x2−208083x−36621194
|
(homogenize, simplify) |
y2z+xyz+yz2=x3+x2z−208083xz2−36621194z3
|
(dehomogenize, simplify) |
y2=x3−269675595x−1704553285050
|
(homogenize, minimize) |
sage:E = EllipticCurve([1, 1, 1, -208083, -36621194])
gp:E = ellinit([1, 1, 1, -208083, -36621194])
magma:E := EllipticCurve([1, 1, 1, -208083, -36621194]);
oscar:E = elliptic_curve([1, 1, 1, -208083, -36621194])
sage:E.short_weierstrass_model()
magma:WeierstrassModel(E);
oscar:short_weierstrass_model(E)
Z
magma:MordellWeilGroup(E);
P | h^(P) | Order |
(1190,36857) | 6.4772043782725117735613891652 | ∞ |
(1190,36857), (1190,−38048)
sage:E.integral_points()
magma:IntegralPoints(E);
Invariants
Conductor: |
N |
= |
1225 | = | 52⋅72 |
sage:E.conductor().factor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
oscar:conductor(E)
|
Discriminant: |
Δ |
= |
−6125 | = | −1⋅53⋅72 |
sage:E.discriminant().factor()
gp:E.disc
magma:Discriminant(E);
oscar:discriminant(E)
|
j-invariant: |
j |
= |
−162677523113838677 | = | −1⋅7⋅1373⋅20833 |
sage:E.j_invariant().factor()
gp:E.j
magma:jInvariant(E);
oscar:j_invariant(E)
|
Endomorphism ring: |
End(E) | = | Z |
Geometric endomorphism ring: |
End(EQ) |
= |
Z
(no potential complex multiplication)
|
sage:E.has_cm()
magma:HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) | = | SU(2) |
Faltings height: |
hFaltings | ≈ | 1.2704847285073684630006342688 |
gp:ellheight(E)
magma:FaltingsHeight(E);
oscar:faltings_height(E)
|
Stable Faltings height: |
hstable | ≈ | 0.54380689222295781849955231159 |
magma:StableFaltingsHeight(E);
oscar:stable_faltings_height(E)
|
abc quality: |
Q | ≈ | 1.0734398000116747 |
|
Szpiro ratio: |
σm | ≈ | 6.799711230731145 |
|
Analytic rank: |
ran | = | 1
|
sage:E.analytic_rank()
gp:ellanalyticrank(E)
magma:AnalyticRank(E);
|
Mordell-Weil rank: |
r | = | 1
|
sage:E.rank()
gp:[lower,upper] = ellrank(E)
magma:Rank(E);
|
Regulator: |
Reg(E/Q) | ≈ | 6.4772043782725117735613891652 |
sage:E.regulator()
gp:G = E.gen \\ if available
matdet(ellheightmatrix(E,G))
magma:Regulator(E);
|
Real period: |
Ω | ≈ | 0.11176617751452526469939232373 |
sage:E.period_lattice().omega()
gp:if(E.disc>0,2,1)*E.omega[1]
magma:(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: |
∏pcp | = | 2
= 2⋅1
|
sage:E.tamagawa_numbers()
gp:gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma:TamagawaNumbers(E);
oscar:tamagawa_numbers(E)
|
Torsion order: |
#E(Q)tor | = | 1 |
sage:E.torsion_order()
gp:elltors(E)[1]
magma:Order(TorsionSubgroup(E));
oscar:prod(torsion_structure(E)[1])
|
Special value: |
L′(E,1) | ≈ | 1.4478647486797316047223462949 |
sage:r = E.rank();
E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp:[r,L1r] = ellanalyticrank(E); L1r/r!
magma:Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
Analytic order of Ш: |
Шan |
≈ |
1
(rounded)
|
sage:E.sha().an_numerical()
magma:MordellWeilShaInformation(E);
|
1.447864749≈L′(E,1)=#E(Q)tor2#Ш(E/Q)⋅ΩE⋅Reg(E/Q)⋅∏pcp≈121⋅0.111766⋅6.477204⋅2≈1.447864749
sage:# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([1, 1, 1, -208083, -36621194]); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
magma:/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve([1, 1, 1, -208083, -36621194]); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular form
1225.2.a.b
q−q2−q3−q4+q6+3q8−2q9+q12+2q13−q16+2q17+2q18−6q19+O(q20)
sage:E.q_eigenform(20)
gp:\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma:ModularForm(E);
For more coefficients, see the Downloads section to the right.
This elliptic curve is not semistable.
There
are 2 primes p
of bad reduction:
sage:E.local_data()
gp:ellglobalred(E)[5]
magma:[LocalInformation(E,p) : p in BadPrimes(E)];
oscar:[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
The ℓ-adic Galois representation has maximal image
for all primes ℓ except those listed in the table below.
sage:rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma:[GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage:gens = [[1, 3256, 0, 1], [1, 0, 148, 1], [1, 28, 0, 1], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [113, 4144, 148, 1], [105, 3034, 37, 783], [75, 74, 1591, 2295], [85, 3108, 148, 3209], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]]
GL(2,Integers(5180)).subgroup(gens)
magma:Gens := [[1, 3256, 0, 1], [1, 0, 148, 1], [1, 28, 0, 1], [1037, 0, 0, 2073], [1925, 3256, 1924, 1925], [4441, 0, 0, 3701], [113, 4144, 148, 1], [105, 3034, 37, 783], [75, 74, 1591, 2295], [85, 3108, 148, 3209], [1, 2590, 0, 1], [2591, 2590, 2590, 2591], [4848, 2923, 4477, 1259], [1, 0, 2590, 1]];
sub<GL(2,Integers(5180))|Gens>;
The image H:=ρE(Gal(Q/Q)) of the adelic Galois representation has
level 5180=22⋅5⋅7⋅37, index 2736, genus 97, and generators
(1032561),(114801),(10281),(1037002073),(1925192432561925),(4441003701),(11314841441),(105373034783),(751591742295),(8514831083209),(1025901),(2591259025902591),(4848447729231259),(1259001).
The torsion field K:=Q(E[5180]) is a degree-61869588480 Galois extension of Q with Gal(K/Q) isomorphic to the projection of H to GL2(Z/5180Z).
The table below list all primes ℓ for which the Serre invariants associated to the mod-ℓ Galois representation are exceptional.
gp:ellisomat(E)
This curve has non-trivial cyclic isogenies of degree d for d=
37.
Its isogeny class 1225.b
consists of 2 curves linked by isogenies of
degree 37.
This elliptic curve is its own minimal quadratic twist.
The number fields K of degree less than 24 such that
E(K)tors is strictly larger than E(Q)tors
(which is trivial)
are as follows:
[K:Q] |
K |
E(K)tors |
Base change curve |
3 |
3.1.980.1 |
Z/2Z |
not in database
|
6 |
6.0.19208000.2 |
Z/2Z⊕Z/2Z |
not in database
|
8 |
8.2.4020286921875.3 |
Z/3Z |
not in database
|
12 |
deg 12 |
Z/4Z |
not in database
|
We only show fields where the torsion growth is primitive.
For fields not in the database, click on the degree shown to reveal the defining polynomial.
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
p-adic regulators
p-adic regulators are not yet computed for curves that are not Γ0-optimal.