Minimal Weierstrass equation
\(y^2+y=x^3+x^2+517x-4340\) 
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \(\left(278, 4658\right)\) |
| \(\hat{h}(P)\) | ≈ | $0.77515645986835787883998033758$ |
Integral points
\( \left(68, 591\right) \), \( \left(68, -592\right) \), \( \left(278, 4658\right) \), \( \left(278, -4659\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 2233 \) | = | \(7 \cdot 11 \cdot 29\) |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | \(-17621717267 \) | = | \(-1 \cdot 7^{3} \cdot 11^{6} \cdot 29 \) |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{15252992000000}{17621717267} \) | = | \(2^{15} \cdot 5^{6} \cdot 7^{-3} \cdot 11^{-6} \cdot 29^{-1} \cdot 31^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | \(0.65238071125574166924238946963\dots\) | ||
| Stable Faltings height: | \(0.65238071125574166924238946963\dots\) | ||
BSD invariants
|
sage: E.rank()
magma: Rank(E);
|
|||
| Analytic rank: | \(1\) | ||
|
sage: E.regulator()
magma: Regulator(E);
|
|||
| Regulator: | \(0.77515645986835787883998033758\dots\) | ||
|
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
|
|||
| Real period: | \(0.66168008298748297825627886096\dots\) | ||
|
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: | \( 6 \) = \( 3\cdot2\cdot1 \) | ||
|
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: | \(1\) (exact) | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 1008 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.0774335441638713604895039724915633614 \)
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | -1 | 1 | 3 | 3 |
| \(11\) | \(2\) | \(I_{6}\) | Non-split multiplicative | 1 | 1 | 6 | 6 |
| \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
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.1.2 |
$p$-adic data
$p$-adic regulators
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ordinary | ss | split | nonsplit | ordinary | ordinary | ordinary | ordinary | split | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 1,2 | 1 | 1,1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 1 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class 2233b
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.812.1 | \(\Z/2\Z\) | Not in database |
| $3$ | 3.1.22707.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.133846832.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $6$ | 6.0.1546823547.1 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| $6$ | 6.0.17802288.1 | \(\Z/6\Z\) | Not in database |
| $9$ | 9.1.7453347531442311744.1 | \(\Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $18$ | 18.0.165709177881786507565918383601203023766699931947.1 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.1499914514460344460786316290785768681472.1 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $18$ | 18.0.552579617605075790794129190535038558318592.1 | \(\Z/2\Z \times \Z/6\Z\) | 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.