Minimal Weierstrass equation
\(y^2=x^3-36x\) 
Mordell-Weil group structure
\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \(\left(-3, 9\right)\) |
| \(\hat{h}(P)\) | ≈ | $0.88862587483961923979418733498$ |
Torsion generators
\( \left(0, 0\right) \), \( \left(6, 0\right) \)
Integral points
\( \left(-6, 0\right) \), \((-3,\pm 9)\), \((-2,\pm 8)\), \( \left(0, 0\right) \), \( \left(6, 0\right) \), \((12,\pm 36)\), \((18,\pm 72)\), \((294,\pm 5040)\)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 576 \) | = | \(2^{6} \cdot 3^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(2985984 \) | = | \(2^{12} \cdot 3^{6} \) |
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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: | $N(\mathrm{U}(1))$ | ||
| Faltings height: | \(-0.068079601017509363137420343386\dots\) | ||
| Stable Faltings height: | \(-1.3105329259115095182522750833\dots\) | ||
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: | \(0.88862587483961923979418733498\dots\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(2.1409010280752311986343110517\dots\) | ||
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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: | \( 16 \) = \( 2^{2}\cdot2^{2} \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(4\) | ||
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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: | 64 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 1.9024600490183925553056020003914001736 \)
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(4\) | \(I_2^{*}\) | Additive | -1 | 6 | 12 | 0 |
| \(3\) | \(4\) | \(I_0^{*}\) | Additive | -1 | 2 | 6 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | Cs |
For all other primes \(p\), the image is 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
\(p\)-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 | add | add | ordinary | ss | ss | ordinary | ordinary | ss | ss | ordinary | ss | ordinary | ordinary | ss | ss |
| $\lambda$-invariant(s) | - | - | 3 | 1,1 | 1,3 | 1 | 1 | 1,3 | 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 576.c
consists of 2 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \times \Z/4\Z\) | 4.4.2304.1-64.1-b2 |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.2.573308928.1 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $8$ | 8.0.2654208000.1 | \(\Z/2\Z \times \Z/10\Z\) | Not in database |
| $16$ | 16.8.118192468620711297024.2 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.0.115422332637413376.2 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.0.29548117155177824256.4 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.0.328683126924509184.1 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/10\Z\) | Not in database |
| $16$ | 16.4.5258930030792146944.1 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/20\Z\) | 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.