Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-789x-7777\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-789xz^2-7777z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12627x-510354\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(37, 94)$ | $0.45251629965890540211752708217$ | $\infty$ |
| $(-77/4, 77/8)$ | $0$ | $2$ |
Integral points
\( \left(-17, 31\right) \), \( \left(-17, -14\right) \), \( \left(-13, 19\right) \), \( \left(-13, -6\right) \), \( \left(37, 94\right) \), \( \left(37, -131\right) \), \( \left(137, 1494\right) \), \( \left(137, -1631\right) \)
Invariants
| Conductor: | $N$ | = | \( 630 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $3986718750$ | = | $2 \cdot 3^{6} \cdot 5^{8} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( \frac{74565301329}{5468750} \) | = | $2^{-1} \cdot 3^{3} \cdot 5^{-8} \cdot 7^{-1} \cdot 23^{3} \cdot 61^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.58829381594869109209532860275$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.038987671614636246397705984289$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9996207719579945$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.906607132255881$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.45251629965890540211752708217$ |
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| Real period: | $\Omega$ | ≈ | $0.90492190390201360011560899734$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2^{3}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6379676457361231357899833054 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.637967646 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.904922 \cdot 0.452516 \cdot 16}{2^2} \\ & \approx 1.637967646\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 512 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 55 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 136 & 3 \\ 141 & 58 \end{array}\right),\left(\begin{array}{rr} 136 & 57 \\ 27 & 70 \end{array}\right),\left(\begin{array}{rr} 161 & 8 \\ 160 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 162 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 64 & 99 \\ 123 & 40 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$3096576$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 | $4$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 630e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 70a4, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.39969909374976.64 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.39033114624.22 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.16647192576.28 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.52509870000.2 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | split | nonsplit | ord | ord | ord | ss | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 2 | 1 | 1 | 1 | 1 | 1,1 | 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 |
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.