Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-7732200x-8275561000\)
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(homogenize, simplify) |
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\(y^2z=x^3-7732200xz^2-8275561000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7732200x-8275561000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1610, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1610, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 705600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $720435416400000000$ | = | $2^{10} \cdot 3^{7} \cdot 5^{8} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{37256083456}{525} \) | = | $2^{11} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-1} \cdot 263^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5667480682974924209546828888$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.33785475724789035577702253588$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9605836908258326$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.395721207507209$ | |||
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: | $\Omega$ | ≈ | $0.090536593980734385239282947241$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4485855036917501638285271559 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.448585504 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.090537 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 1.448585504\end{aligned}$$
Modular invariants
Modular form 705600.2.a.bwv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18874368 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | 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 | 16.24.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 195 & 1664 \\ 526 & 309 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 659 & 1664 \\ 592 & 315 \end{array}\right),\left(\begin{array}{rr} 1184 & 1675 \\ 765 & 14 \end{array}\right),\left(\begin{array}{rr} 1112 & 1679 \\ 481 & 1670 \end{array}\right),\left(\begin{array}{rr} 1679 & 1664 \\ 1260 & 1259 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1582 & 1667 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1676 & 1677 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$5945425920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
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 | $2$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 78400 = 2^{6} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 4, 8 and 8.
Its isogeny class 705600.bwv
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840.d4, its twist by $-840$.
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 $0$.