Minimal Weierstrass equation
\(y^2+xy=x^3-x^2-286917x+58100741\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(254, 1123\right) \) |
| \(\hat{h}(P)\) | ≈ | $1.2953666381816699183949433575$ |
Torsion generators
\( \left(274, -137\right) \)
Integral points
\( \left(254, 1123\right) \), \( \left(254, -1377\right) \), \( \left(274, -137\right) \), \( \left(899, 22488\right) \), \( \left(899, -23387\right) \), \( \left(3379, 192373\right) \), \( \left(3379, -195752\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 72450 \) | = | \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(56942303906250000 \) | = | \(2^{4} \cdot 3^{9} \cdot 5^{11} \cdot 7 \cdot 23^{2} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{8493409990827}{185150000} \) | = | \(2^{-4} \cdot 3^{6} \cdot 5^{-5} \cdot 7^{-1} \cdot 23^{-2} \cdot 2267^{3}\) |
| 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: | \(1.2953666381816699183949433575\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(0.35232005715906621146647104156\) | ||
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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: | \( 32 \) = \( 2\cdot2\cdot2^{2}\cdot1\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(2\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 72450.2.a.j
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: | 737280 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.6510691840489070847976718458087404422 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | 1 | 1 | 4 | 4 |
| \(3\) | \(2\) | \(III^{*}\) | Additive | 1 | 2 | 9 | 0 |
| \(5\) | \(4\) | \(I_5^{*}\) | Additive | 1 | 2 | 11 | 5 |
| \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(23\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-adic regulators
Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | add | nonsplit | ordinary | ordinary | ordinary | ordinary | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 4 | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class 72450b
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $4$ | 4.0.7998480.3 | \(\Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | 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.