Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-3577352115x-82343521821135\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-3577352115xz^2-82343521821135z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4636248341067x-3841749810361750266\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-139549/4, 139545/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 424830 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $931838076410362593504123840$ | = | $2^{6} \cdot 3 \cdot 5 \cdot 7^{8} \cdot 17^{14} $ |
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| j-invariant: | $j$ | = | \( \frac{1782900110862842086081}{328139630024640} \) | = | $2^{-6} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 527207^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1776470733658426599728769171$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7880853268100779672954332364$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9994946520793547$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.988471960889188$ | |||
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: | $\Omega$ | ≈ | $0.019521533427782338730477995158$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2 \cdot 3 )\cdot1\cdot1\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.8740672090671045181258875352 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.874067209 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.019522 \cdot 1.000000 \cdot 96}{2^2} \\ & \approx 1.874067209\end{aligned}$$
Modular invariants
Modular form 424830.2.a.fi
For more coefficients, see the Downloads section to the right.
| Modular degree: | 509607936 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $17$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
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 | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28560 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 19048 & 4081 \\ 17759 & 8170 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28545 & 16 \\ 28544 & 17 \end{array}\right),\left(\begin{array}{rr} 3578 & 9191 \\ 427 & 1926 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 28462 & 28547 \end{array}\right),\left(\begin{array}{rr} 22856 & 4081 \\ 19663 & 8170 \end{array}\right),\left(\begin{array}{rr} 20413 & 8176 \\ 27804 & 15625 \end{array}\right),\left(\begin{array}{rr} 8159 & 0 \\ 0 & 28559 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 28556 & 28557 \end{array}\right),\left(\begin{array}{rr} 20159 & 20384 \\ 14392 & 20271 \end{array}\right)$.
The torsion field $K:=\Q(E[28560])$ is a degree-$465740884869120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28560\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 212415 = 3 \cdot 5 \cdot 7^{2} \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 70805 = 5 \cdot 7^{2} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 84966 = 2 \cdot 3 \cdot 7^{2} \cdot 17^{2} \) |
| $7$ | additive | $32$ | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 424830.fi
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3570.t2, its twist by $-119$.
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 $0$.