Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-4219\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-4219z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-270000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
| Conductor: | \( 225 \) | = | $3^{2} \cdot 5^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
| Discriminant: | $-7688671875 $ | = | $-1 \cdot 3^{9} \cdot 5^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
| j-invariant: | \( 0 \) | = | $0$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $0.57580039636244460329040428732\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
| Stable Faltings height: | $-1.3211174284280379149898691959\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
| Real period: | $0.60418195389101983687716770297\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
| Tamagawa product: | $ 2 $ = $ 2\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
| Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
| Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
| Special value: | $ L(E,1) $ ≈ $ 1.2083639077820396737543354059 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 1.208363908 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.604182 \cdot 1.000000 \cdot 2}{1^2} \approx 1.208363908$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 120 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
| $5$ | $1$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 27.648.18.4 |
| $5$ | 5Ns.2.1 | 5.30.0.2 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 225.d
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 225.c2, its twist by $-15$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $-675$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-5625.1-CMb1 |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | Not in database |
| $3$ | 3.1.675.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.1366875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $8$ | 8.4.56953125.1 | \(\Z/5\Z\) | Not in database |
| $9$ | 9.1.59049000000.1 | \(\Z/6\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/21\Z\) | Not in database |
| $12$ | 12.2.8062156800000000.1 | \(\Z/4\Z\) | Not in database |
| $16$ | 16.0.3243658447265625.1 | \(\Z/5\Z \oplus \Z/15\Z\) | Not in database |
| $18$ | 18.0.150799169014635498046875.2 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.10460353203000000000000.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 |
|---|---|---|---|
| Reduction type | ss | add | add |
| $\lambda$-invariant(s) | ? | - | - |
| $\mu$-invariant(s) | ? | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.