Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+73500x+2846900000\)
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(homogenize, simplify) |
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\(y^2z=x^3+73500xz^2+2846900000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+73500x+2846900000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1400, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1400, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 705600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-3501316123704000000000$ | = | $-1 \cdot 2^{12} \cdot 3^{12} \cdot 5^{9} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{64}{5103} \) | = | $2^{6} \cdot 3^{-6} \cdot 7^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8128416869792237127273442690$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.60964514676800837589075634256$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2117021900209282$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.237244374693263$ | |||
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: | $\Omega$ | ≈ | $0.11131784291175469689256539513$ |
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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)$ | ≈ | $1.7810854865880751502810463220 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.781085487 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.111318 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 1.781085487\end{aligned}$$
Modular invariants
Modular form 705600.2.a.hi
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35389440 |
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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 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $7$ | $4$ | $I_{1}^{*}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 281 & 4 \\ 142 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 62 & 1 \\ 179 & 0 \end{array}\right),\left(\begin{array}{rr} 417 & 4 \\ 416 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 109 & 316 \\ 104 & 315 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 88 & 1 \\ 83 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$371589120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 | $2$ | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 78400 = 2^{6} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | 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.
Its isogeny class 705600.hi
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 16800.y2, 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 $0$.