Minimal Weierstrass equation
\(y^2=x^3-108x-540\)
Mordell-Weil group structure
trivial
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 216 \) | = | $2^{3} \cdot 3^{3}$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-45349632 $ | = | $-1 \cdot 2^{8} \cdot 3^{11} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -3072 \) | = | $-1 \cdot 2^{10} \cdot 3$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.18171061573467243251615648170\dots$ | ||
| Stable Faltings height: | $-1.2874487692510583242076397331\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.72841724621303281407275386739\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 2 $ = $ 2\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 1.4568344924260656281455077348 $ | ||
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: | 72 | ||
| $ \Gamma_0(N) $-optimal: | yes | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{1}^{*}$ | Additive | -1 | 3 | 8 | 0 |
| $3$ | $1$ | $II^{*}$ | Additive | 1 | 3 | 11 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2G | 4.4.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 |
| $\mu$-invariant(s) | - | - | 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 no rational isogenies. Its isogeny class 216.d consists of this curve only.
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 |
|---|---|---|---|
| $3$ | 3.1.108.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $8$ | 8.2.181398528.1 | \(\Z/3\Z\) | Not in database |
| $12$ | 12.2.15045919506432.1 | \(\Z/4\Z\) | Not in database |
We only show fields where the torsion growth is primitive.