Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+58564\) | (homogenize, simplify) |
\(y^2z=x^3+58564z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+58564\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(20, 258\right)\) |
$\hat{h}(P)$ | ≈ | $2.6982393721493257909895649903$ |
Torsion generators
\( \left(0, 242\right) \)
Integral points
\((0,\pm 242)\), \((20,\pm 258)\), \((605,\pm 14883)\)
Invariants
Conductor: | \( 4356 \) | = | $2^{2} \cdot 3^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1481648585472 $ | = | $-1 \cdot 2^{8} \cdot 3^{3} \cdot 11^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( 0 \) | = | $0$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
Faltings height: | $1.0142306126445334101783925797\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.3211174284280379149898691958\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $\dots$ | |||
Szpiro ratio: | $4.23410741777193\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.6982393721493257909895649903\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.67502608109450615331078692939\dots$ | 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);
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Tamagawa product: | $ 18 $ = $ 3\cdot2\cdot3 $ | 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)
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Torsion order: | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.6427638984737203185149136372 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.642763898 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.675026 \cdot 2.698239 \cdot 18}{3^2} \approx 3.642763898$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 3168 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
$3$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
$11$ | $3$ | $IV^{*}$ | 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.1.1 | 27.648.18.1 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 4356.e
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4356.f1, its twist by $-11$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $484$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.1452.1 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.6324912.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.0.512317872.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$9$ | 9.3.1626877698164928.25 | \(\Z/9\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/21\Z\) | Not in database |
$18$ | 18.0.7940193134359243721936067735552.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.134467867946269093836238848.1 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | add | ss | ord | add | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
$\lambda$-invariant(s) | - | - | 1,3 | 1 | - | 3 | 1,1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.