Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+2547x-102735\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+2547xz^2-102735z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+40749x-6534290\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(192, 2631)$ | $3.9661320246363729029285178722$ | $\infty$ |
| $(30, -15)$ | $0$ | $2$ |
Integral points
\( \left(30, -15\right) \), \( \left(192, 2631\right) \), \( \left(192, -2823\right) \)
Invariants
| Conductor: | $N$ | = | \( 410958 \) | = | $2 \cdot 3^{2} \cdot 17^{2} \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $-5560426945116$ | = | $-1 \cdot 2^{2} \cdot 3^{6} \cdot 17^{6} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{103823}{316} \) | = | $2^{-2} \cdot 47^{3} \cdot 79^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1288734288028031737247879567$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.83703938755935971209760197070$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8000937907084497$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.830715664297125$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9661320246363729029285178722$ |
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| Real period: | $\Omega$ | ≈ | $0.38973415073530132498125539930$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0914741926514759703431850025 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.091474193 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.389734 \cdot 3.966132 \cdot 8}{2^2} \\ & \approx 3.091474193\end{aligned}$$
Modular invariants
Modular form 410958.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $79$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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 | 8.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 632 = 2^{3} \cdot 79 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 322 & 1 \\ 471 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 81 & 554 \\ 552 & 79 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 317 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 629 & 4 \\ 628 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[632])$ is a degree-$4921712640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/632\Z)$.
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 | $4$ | \( 205479 = 3^{2} \cdot 17^{2} \cdot 79 \) |
| $3$ | additive | $2$ | \( 45662 = 2 \cdot 17^{2} \cdot 79 \) |
| $17$ | additive | $146$ | \( 1422 = 2 \cdot 3^{2} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 410958.h
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 158.e2, its twist by $-51$.
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.