Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-9158x+150518\) 
Mordell-Weil group structure
\(\Z^2\)
Infinite order Mordell-Weil generators and heights
| \(P\) | = | \(\left(114, 703\right)\) | \(\left(-\frac{389}{4}, \frac{3089}{8}\right)\) |
| \(\hat{h}(P)\) | ≈ | $0.38763373764020226581950487268$ | $3.1773844652955484372361827125$ |
Integral points
\( \left(-12, 514\right) \), \( \left(-12, -503\right) \), \( \left(114, 703\right) \), \( \left(114, -818\right) \), \( \left(153, 1444\right) \), \( \left(153, -1598\right) \), \( \left(3156, 175618\right) \), \( \left(3156, -178775\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 439569 \) | = | \(3^{2} \cdot 13^{2} \cdot 17^{2}\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(39659760930231 \) | = | \(3^{7} \cdot 13^{7} \cdot 17^{2} \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{83521}{39} \) | = | \(3^{-1} \cdot 13^{-1} \cdot 17^{4}\) |
| 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);
|
|||
| Analytic rank: | \(2\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(1.2140971972541388716190173942\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.57758809570822295728397790215\) | ||
|
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
|
|||
| Tamagawa product: | \( 16 \) = \( 2^{2}\cdot2^{2}\cdot1 \) | ||
|
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
|
|||
| Torsion order: | \(1\) | ||
|
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
|||
| Analytic order of Ш: | \(1\) (rounded) | ||
Modular invariants
Modular form 439569.2.a.s
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 774144 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L^{(2)}(E,1)/2! \) ≈ \( 11.219969410667340947094289755865961644 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(4\) | \(I_1^{*}\) | Additive | -1 | 2 | 7 | 1 |
| \(13\) | \(4\) | \(I_1^{*}\) | Additive | 1 | 2 | 7 | 1 |
| \(17\) | \(1\) | \(II\) | 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 \( p \) .
$p$-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 no rational isogenies. Its isogeny class 439569.s consists of this curve only.