Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1430827x-657916211\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1430827xz^2-657916211z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1854352467x-30667923456786\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3705, 210083)$ | $8.3883900595896567124340314199$ | $\infty$ |
| $(-2869/4, 2869/8)$ | $0$ | $2$ |
Integral points
\( \left(3705, 210083\right) \), \( \left(3705, -213788\right) \)
Invariants
| Conductor: | $N$ | = | \( 3630 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $820259640539800560$ | = | $2^{4} \cdot 3^{3} \cdot 5 \cdot 11^{14} $ |
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| j-invariant: | $j$ | = | \( \frac{182864522286982801}{463015182960} \) | = | $2^{-4} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-8} \cdot 567601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3141336696921428681209122011$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1151860332929575960899404121$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0750149448569075$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.6042422015747455$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.3883900595896567124340314199$ |
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| Real period: | $\Omega$ | ≈ | $0.13806030160159093610171773681$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.3162073231574707410803964607 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.316207323 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.138060 \cdot 8.388390 \cdot 8}{2^2} \\ & \approx 2.316207323\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 92160 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
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 | 16.24.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1439 & 2624 \\ 952 & 2511 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 668 & 789 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2542 & 2627 \end{array}\right),\left(\begin{array}{rr} 338 & 1987 \\ 2335 & 1342 \end{array}\right),\left(\begin{array}{rr} 896 & 5 \\ 2595 & 2626 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 536 & 1 \\ 1663 & 10 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2636 & 2637 \end{array}\right),\left(\begin{array}{rr} 2625 & 16 \\ 2624 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2640])$ is a degree-$38928384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2640\Z)$.
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 | $4$ | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 3630d
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330c4, its twist by $-11$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-165}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | 2.0.11.1-9900.5-f7 |
| $4$ | \(\Q(\sqrt{-11}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.10929447936000000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.10929447936000000.79 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.189747360000.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.8433216000000.18 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | ss | add | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 3 | 2 | 1,3 | - | 1 | 1 | 3 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0,0 | - | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.