Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-833x\)
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(homogenize, simplify) |
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\(y^2z=x^3-833xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-833x\)
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(homogenize, minimize) |
comment: Define the curve
sage: E = EllipticCurve([0, 0, 0, -833, 0])
gp: E = ellinit([0, 0, 0, -833, 0])
magma: E := EllipticCurve([0, 0, 0, -833, 0]);
oscar: E = EllipticCurve([0, 0, 0, -833, 0])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroup(E);
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{729}{25}, \frac{2808}{125}\right)\)
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| $\hat{h}(P)$ | ≈ | $6.5888909426376150630116747419$ |
sage: E.gens()
magma: Generators(E);
gp: E.gen
Torsion generators
\( \left(0, 0\right) \)
comment: Torsion subgroup
sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);
oscar: torsion_structure(E)
Integral points
\( \left(0, 0\right) \)
comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
| Conductor: | \( 453152 \) | = | $2^{5} \cdot 7^{2} \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $36992610368 $ | = | $2^{6} \cdot 7^{6} \cdot 17^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $0.71729907491017380907140100362\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.3105329259115095182522750833\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $\dots$ | |||
| Szpiro ratio: | $2.440782073356161\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $6.5888909426376150630116747419\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.97613742873619808195507039729\dots$ | comment: Real Period
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);
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| Tamagawa product: | $ 8 $ = $ 2\cdot2\cdot2 $ | comment: Tamagawa numbers
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)
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| Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 12.863326125939011955686444189 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 12.863326126 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.976137 \cdot 6.588891 \cdot 8}{2^2} \approx 12.863326126$
# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve(%s); 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)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */
E := EllipticCurve(%s); 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 invariants
Modular form 453152.2.a.bc
\( q + 4 q^{5} - 3 q^{9} + 6 q^{13} + O(q^{20}) \)
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comment: q-expansion of modular form
sage: E.q_eigenform(20)
\\ 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.
| Modular degree: | 393216 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | Additive | -1 | 5 | 6 | 0 |
| $7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
comment: Local data
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)]
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
comment: mod p Galois image
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
Isogenies
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 453152.bc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 9248.c1, its twist by $28$.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $240737$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.