Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-10699563x-11706686138\)
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(homogenize, simplify) |
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\(y^2z=x^3-10699563xz^2-11706686138z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10699563x-11706686138\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2418, 5242\right) \) | $7.8730871067997247452046491857$ | $\infty$ |
| \( \left(-2422, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-2418:5242:1]\) | $7.8730871067997247452046491857$ | $\infty$ |
| \([-2422:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2418, 5242\right) \) | $7.8730871067997247452046491857$ | $\infty$ |
| \( \left(-2422, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-2422, 0\right) \), \((-2418,\pm 5242)\)
\([-2422:0:1]\), \([-2418:\pm 5242:1]\)
\( \left(-2422, 0\right) \), \((-2418,\pm 5242)\)
Invariants
| Conductor: | $N$ | = | \( 405720 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ |
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| Minimal Discriminant: | $\Delta$ | = | $19189058079228307200000$ | = | $2^{11} \cdot 3^{10} \cdot 5^{5} \cdot 7^{3} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{264527137402013054}{37471584403125} \) | = | $2 \cdot 3^{-4} \cdot 5^{-5} \cdot 23^{-6} \cdot 137^{3} \cdot 3719^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0016256692034603456162957003$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3304570720922939733432054513$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9801433737893019$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.659551860198044$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.8730871067997247452046491857$ |
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| Real period: | $\Omega$ | ≈ | $0.084262339806186282592661790546$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 1\cdot2^{2}\cdot1\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.9608568934023492649544922423 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.960856893 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.084262 \cdot 7.873087 \cdot 48}{2^2} \\ & \approx 7.960856893\end{aligned}$$
Modular invariants
Modular form 405720.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26542080 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 3 | 11 | 0 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $23$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6440 = 2^{3} \cdot 5 \cdot 7 \cdot 23 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 5636 & 809 \\ 4025 & 2416 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 281 & 4 \\ 562 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4604 & 1 \\ 3679 & 0 \end{array}\right),\left(\begin{array}{rr} 6437 & 4 \\ 6436 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 3219 & 0 \end{array}\right),\left(\begin{array}{rr} 5154 & 1 \\ 3863 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[6440])$ is a degree-$33092240670720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6440\Z)$.
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 | $4$ | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| $3$ | additive | $8$ | \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 81144 = 2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 23 \) |
| $7$ | additive | $20$ | \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 17640 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 405720p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 135240ck2, its twist by $-3$.
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.