Minimal Weierstrass equation
\(y^2+xy=x^3+x^2+154227x+8862813\)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
\(P\) | = | \( \left(\frac{2225991}{676}, \frac{3316756923}{17576}\right) \) |
\(\hat{h}(P)\) | ≈ | $14.125930840273181391561017988$ |
Integral points
None
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 424830 \) | = | \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}\) |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | \(-268220750584872960 \) | = | \(-1 \cdot 2^{33} \cdot 3^{2} \cdot 5 \cdot 7^{4} \cdot 17^{2} \) |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{584669638645799}{386547056640} \) | = | \(2^{-33} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{2} \cdot 17 \cdot 8887^{3}\) |
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | \(1\) | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | \(14.125930840273181391561017988\) | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | \(0.19427088083910717512482196971\) | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | \( 2 \) = \( 1\cdot2\cdot1\cdot1\cdot1 \) | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | \(1\) | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | \(1\) (exact) |
Modular invariants
Modular form 424830.2.a.t
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 6928416 | ||
\( \Gamma_0(N) \)-optimal: | no | ||
Manin constant: | 1 |
Special L-value
\( L'(E,1) \) ≈ \( 5.4885140540243606254568310309298675484 \)
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|
\(2\) | \(1\) | \(I_{33}\) | Non-split multiplicative | 1 | 1 | 33 | 33 |
\(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
\(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
\(7\) | \(1\) | \(IV\) | Additive | 1 | 2 | 4 | 0 |
\(17\) | \(1\) | \(II\) | Additive | 1 | 2 | 2 | 0 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
prime | Image of Galois representation |
---|---|
\(3\) | B |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class 424830.t
consists of 2 curves linked by isogenies of
degree 3.