Minimal Weierstrass equation
\(y^2=x^3-54x\)
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \( \left(-2, 10\right) \) |
| \(\hat{h}(P)\) | ≈ | $1.7615065287953594805667428175$ |
Torsion generators
\( \left(0, 0\right) \)
Integral points
\((-2,\pm 10)\), \( \left(0, 0\right) \), \((27,\pm 135)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 2304 \) | = | \(2^{8} \cdot 3^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(10077696 \) | = | \(2^{9} \cdot 3^{9} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( 1728 \) | = | \(2^{6} \cdot 3^{3}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[\sqrt{-1}]\) | (potential complex multiplication) | |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
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: | \(1.7615065287953594805667428175\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(1.9345224584991555734883335751\) | ||
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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: | \( 4 \) = \( 2\cdot2 \) | ||
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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
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: | 384 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.4076739407475124029238699904600992021 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(III\) | Additive | 1 | 8 | 9 | 0 |
| \(3\) | \(2\) | \(III^{*}\) | Additive | 1 | 2 | 9 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
$p$-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 | add | add | ordinary | ss | ss | ordinary | ordinary | ss | ss | ordinary | ss | ordinary | ordinary | ss | ss |
| $\lambda$-invariant(s) | - | - | 1 | 3,1 | 1,1 | 1 | 1 | 5,1 | 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 | 0,0 | 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 2304.m
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $4$ | 4.0.55296.1 | \(\Z/4\Z\) | Not in database |
| $8$ | 8.4.12230590464.5 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.12230590464.5 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.2.36691771392.3 | \(\Z/6\Z\) | Not in database |
| $8$ | 8.0.1528823808000.20 | \(\Z/10\Z\) | Not in database |
| $16$ | 16.0.149587343098087735296.14 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | 16.0.1346286087882789617664.1 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/10\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/10\Z\) | 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.