Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2380x-44688\)
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(homogenize, simplify) |
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\(y^2z=x^3-2380xz^2-44688z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2380x-44688\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-28, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-28:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-28, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-28, 0\right) \)
\([-28:0:1]\)
\( \left(-28, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 3136 \) | = | $2^{6} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $89915392$ | = | $2^{18} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( 16581375 \) | = | $3^{3} \cdot 5^{3} \cdot 17^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-7}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.58715597206426820171015253319$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.93904233603947808869203383486$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1980441775334598$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.339766189018317$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.68352890859448887243602360704$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3670578171889777448720472141 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.367057817 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.683529 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 1.367057817\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1024 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | 1 | 6 | 18 | 0 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 7 \) |
| $7$ | additive | $20$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 3136d
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49a2, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.21952.1 | \(\Z/4\Z\) | not in database |
| $6$ | 6.0.8605184.1 | \(\Z/14\Z\) | not in database |
| $8$ | 8.0.7710244864.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.7710244864.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1927561216.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1927561216.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.1053894094848.2 | \(\Z/6\Z\) | not in database |
| $12$ | 12.0.4739148267126784.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | 16.0.59447875862838378496.4 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.59447875862838378496.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $20$ | 20.0.12020004674398148709330936922112.1 | \(\Z/22\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | add | ss | ss | add |
| $\lambda$-invariant(s) | - | 0,0 | 0,0 | - |
| $\mu$-invariant(s) | - | 0,0 | 0,0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.