Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5856400\)
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(homogenize, simplify) |
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\(y^2z=x^3-5856400z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5856400\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-14816485854720000$ | = | $-1 \cdot 2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( 0 \) | = | $0$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7817589769758819715177230646$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3211174284280379149898691958$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.441742599614708$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.18089501224293954925563354069$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot2\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0853700734576372955338012441 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.085370073 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.180895 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 1.085370073\end{aligned}$$
Modular invariants
Modular form 435600.2.a.a
\( q - 5 q^{7} - 7 q^{13} - 8 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 4333824 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $11$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $14$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 435600.a
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 27225.w1, its twist by $44$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by $-75625$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.