Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-64431588x+198703499488\)
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(homogenize, simplify) |
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\(y^2z=x^3-64431588xz^2+198703499488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-64431588x+198703499488\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4472, 0)$ | $0$ | $2$ |
Integral points
\( \left(4472, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 486720 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $62233756518821921280000$ | = | $2^{12} \cdot 3^{18} \cdot 5^{4} \cdot 13^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{2052450196928704}{4317958125} \) | = | $2^{6} \cdot 3^{-12} \cdot 5^{-4} \cdot 13^{-1} \cdot 31771^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2588467452938328482230440760$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73391874166906432508144561530$ |
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| $abc$ quality: | $Q$ | ≈ | $0.997177538945588$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.006088017622679$ | |||
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: | $\Omega$ | ≈ | $0.11086180219698563720671930020$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7737888351517701953075088032 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.773788835 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.110862 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 1.773788835\end{aligned}$$
Modular invariants
Modular form 486720.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49545216 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 6 | 12 | 0 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 8.12.0.23 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 104 = 2^{3} \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 96 \\ 100 & 71 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 71 & 98 \\ 6 & 5 \end{array}\right),\left(\begin{array}{rr} 97 & 8 \\ 96 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 45 & 98 \\ 6 & 5 \end{array}\right),\left(\begin{array}{rr} 88 & 101 \\ 3 & 102 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right)$.
The torsion field $K:=\Q(E[104])$ is a degree-$838656$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/104\Z)$.
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 | $2$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 54080 = 2^{6} \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 97344 = 2^{6} \cdot 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 486720ce
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 6240y1, its twist by $-312$.
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 $0$.