Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-9927742x-10308063084\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-9927742xz^2-10308063084z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-158843875x-659874881250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2356, 1178)$ | $0$ | $2$ |
Integral points
\( \left(-2356, 1178\right) \)
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $16697880267112000000000$ | = | $2^{12} \cdot 5^{9} \cdot 11^{2} \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{93144487437}{14372864} \) | = | $2^{-12} \cdot 3^{3} \cdot 11^{-2} \cdot 29^{-1} \cdot 1511^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9875575379488313487005301865$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.096831188630019054158324670402$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9690593234392751$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.595506786276608$ | |||
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: | $\Omega$ | ≈ | $0.085935121178034935091385853199$ |
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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(E,1)$ | ≈ | $2.7499238776971179229243473024 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.749923878 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.085935 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.749923878\end{aligned}$$
Modular invariants
Modular form 462550.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 38707200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $29$ | $4$ | $I_{1}^{*}$ | 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 \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 577 & 4 \\ 576 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 437 & 146 \\ 144 & 435 \end{array}\right),\left(\begin{array}{rr} 468 & 1 \\ 463 & 0 \end{array}\right),\left(\begin{array}{rr} 62 & 1 \\ 259 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$2619187200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
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 | $4$ | \( 4205 = 5 \cdot 29^{2} \) |
| $3$ | good | $2$ | \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \) |
| $5$ | additive | $14$ | \( 18502 = 2 \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462550.q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 15950.e2, its twist by $145$.
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 $0$.