Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-497100x-69874000\)
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(homogenize, simplify) |
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\(y^2z=x^3-497100xz^2-69874000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-497100x-69874000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(980, 19600\right) \) | $4.8984706571411059963556463882$ | $\infty$ |
| \( \left(-620, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([980:19600:1]\) | $4.8984706571411059963556463882$ | $\infty$ |
| \([-620:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(980, 19600\right) \) | $4.8984706571411059963556463882$ | $\infty$ |
| \( \left(-620, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-620, 0\right) \), \((980,\pm 19600)\)
\([-620:0:1]\), \([980:\pm 19600:1]\)
\( \left(-620, 0\right) \), \((980,\pm 19600)\)
Invariants
| Conductor: | $N$ | = | \( 417600 \) | = | $2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $5752419420672000000$ | = | $2^{15} \cdot 3^{18} \cdot 5^{6} \cdot 29 $ |
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| j-invariant: | $j$ | = | \( \frac{36396323144}{15411789} \) | = | $2^{3} \cdot 3^{-12} \cdot 29^{-1} \cdot 1657^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2964304918138517456597896018$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.075971415562815075890247164902$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0545866958122545$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9377331663471353$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8984706571411059963556463882$ |
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| Real period: | $\Omega$ | ≈ | $0.18675045108979700470490687629$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.3183328389698872648349559411 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.318332839 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.186750 \cdot 4.898471 \cdot 32}{2^2} \\ & \approx 7.318332839\end{aligned}$$
Modular invariants
Modular form 417600.2.a.he
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5505024 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 6 | 15 | 0 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $29$ | $1$ | $I_{1}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1304 & 1645 \\ 3215 & 1794 \end{array}\right),\left(\begin{array}{rr} 1516 & 2785 \\ 2975 & 6 \end{array}\right),\left(\begin{array}{rr} 3401 & 2700 \\ 1310 & 2351 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2783 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3474 & 3475 \end{array}\right),\left(\begin{array}{rr} 3473 & 8 \\ 3472 & 9 \end{array}\right),\left(\begin{array}{rr} 1159 & 2080 \\ 460 & 1359 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$502883942400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $4$ | \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \) |
| $3$ | additive | $6$ | \( 46400 = 2^{6} \cdot 5^{2} \cdot 29 \) |
| $5$ | additive | $14$ | \( 16704 = 2^{6} \cdot 3^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 14400 = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 417600.he
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2784.b2, its twist by $-120$.
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.