Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-2218576x+1271222298\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-2218576xz^2+1271222298z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2875273875x+59318773368750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(766, 4242)$ | $1.4765844611262246649804094686$ | $\infty$ |
| $(911, 2067)$ | $2.1547365313776320277569940465$ | $\infty$ |
| $(3383/4, -3387/8)$ | $0$ | $2$ |
Integral points
\( \left(-1467, 37737\right) \), \( \left(-1467, -36271\right) \), \( \left(766, 4242\right) \), \( \left(766, -5009\right) \), \( \left(877, 61\right) \), \( \left(877, -939\right) \), \( \left(911, 2067\right) \), \( \left(911, -2979\right) \), \( \left(918, 2439\right) \), \( \left(918, -3358\right) \), \( \left(1627, 43561\right) \), \( \left(1627, -45189\right) \), \( \left(1752, 51686\right) \), \( \left(1752, -53439\right) \), \( \left(598021, 462160161\right) \), \( \left(598021, -462758183\right) \)
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $562293920632812500$ | = | $2^{2} \cdot 5^{9} \cdot 11^{2} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1039509197}{484} \) | = | $2^{-2} \cdot 11^{-2} \cdot 1013^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3621166817917611705817009992$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.52860966752705112396050451690$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9312136896197799$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.250886462372206$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0360734997109201130062987441$ |
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| Real period: | $\Omega$ | ≈ | $0.28705054362112856923056413509$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.9720523885257756033788384072 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.972052389 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.287051 \cdot 3.036073 \cdot 32}{2^2} \\ & \approx 6.972052389\end{aligned}$$
Modular invariants
Modular form 462550.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10752000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $29$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 \( 220 = 2^{2} \cdot 5 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 57 & 166 \\ 164 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 217 & 4 \\ 216 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 101 & 4 \\ 202 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48 & 1 \\ 43 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[220])$ is a degree-$50688000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/220\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 4205 = 5 \cdot 29^{2} \) |
| $5$ | additive | $14$ | \( 18502 = 2 \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462550.e
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 550.c1, its twist by $145$.
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.