Minimal Weierstrass equation
\(y^2=x^3-x^2-51866085x-2718075836595\)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(39692, 7599647\right) \) |
| \(\hat{h}(P)\) | ≈ | $8.4052688058246229872315506891$ |
Integral points
\((39692,\pm 7599647)\)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 203280 \) | = | \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-3182699496026549496328826880 \) | = | \(-1 \cdot 2^{12} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{21} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{2126464142970105856}{438611057788643355} \) | = | \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-15} \cdot 179^{3} \cdot 449^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(8.4052688058246229872315506891\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.020000894867476545796315619116\) | ||
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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: | \( 2 \) = \( 1\cdot1\cdot1\cdot1\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(25\) = $5^2$ (exact) | ||
Modular invariants
Modular form 203280.2.a.ds
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 144000000 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 8.4056448859089208561924128312077918293 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \(II^{*}\) | Additive | -1 | 4 | 12 | 0 |
| \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(11\) | \(2\) | \(I_{15}^{*}\) | Additive | -1 | 2 | 21 | 15 |
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 |
|---|---|
| \(5\) | B.4.2 |
$p$-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 \(d=\)
5.
Its isogeny class 203280.ds
consists of 2 curves linked by isogenies of
degree 5.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.4620.1 | \(\Z/2\Z\) | Not in database |
| $4$ | 4.0.242000.2 | \(\Z/5\Z\) | Not in database |
| $6$ | 6.0.24652782000.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/3\Z\) | Not in database |
| $10$ | 10.2.23794567667767230468750000000000.1 | \(\Z/5\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/10\Z\) | 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.