Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-86583x-1209418\)
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(homogenize, simplify) |
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\(y^2z=x^3-86583xz^2-1209418z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-86583x-1209418\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-131, 2808)$ | $2.2093468202795434981990355400$ | $\infty$ |
| $(337, 2808)$ | $2.5231818187637291139678151040$ | $\infty$ |
| $(-14, 0)$ | $0$ | $2$ |
| $(301, 0)$ | $0$ | $2$ |
Integral points
\( \left(-287, 0\right) \), \((-259,\pm 1960)\), \((-206,\pm 2808)\), \((-131,\pm 2808)\), \((-62,\pm 1980)\), \( \left(-14, 0\right) \), \( \left(301, 0\right) \), \((337,\pm 2808)\), \((721,\pm 17640)\), \((742,\pm 18522)\), \((5446,\pm 401310)\), \((2963821,\pm 5102440560)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $40909204684857600$ | = | $2^{8} \cdot 3^{8} \cdot 5^{2} \cdot 7^{8} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3269383504}{1863225} \) | = | $2^{4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 19^{3} \cdot 31^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8773248187797676021014425734$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.10703452045524076909367783109$ |
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| $abc$ quality: | $Q$ | ≈ | $0.879166282565753$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.507218944218687$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.5728423099226602582159973516$ |
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| Real period: | $\Omega$ | ≈ | $0.30103778634126033635223106141$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.421088900864268131885467905 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.421088901 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.301038 \cdot 5.572842 \cdot 128}{4^2} \\ & \approx 13.421088901\end{aligned}$$
Modular invariants
Modular form 458640.2.a.w
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3932160 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3277 & 4 \\ 1094 & 9 \end{array}\right),\left(\begin{array}{rr} 1683 & 2 \\ 2518 & 5459 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2729 & 5458 \\ 0 & 5459 \end{array}\right),\left(\begin{array}{rr} 5457 & 4 \\ 5456 & 5 \end{array}\right),\left(\begin{array}{rr} 3119 & 5458 \\ 0 & 5459 \end{array}\right),\left(\begin{array}{rr} 1819 & 5456 \\ 3638 & 5451 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$2434651914240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 458640w
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 10920k2, its twist by $-84$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.