Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+249x+16126\)
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(homogenize, simplify) |
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\(y^2z=x^3+249xz^2+16126z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+249x+16126\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1, 126)$ | $0.69440500528088963971638506800$ | $\infty$ |
| $(-22, 0)$ | $0$ | $2$ |
Integral points
\( \left(-22, 0\right) \), \((-1,\pm 126)\), \((3,\pm 130)\), \((59,\pm 486)\), \((4094,\pm 261954)\)
Invariants
| Conductor: | $N$ | = | \( 4284 \) | = | $2^{2} \cdot 3^{2} \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-113328730368$ | = | $-1 \cdot 2^{8} \cdot 3^{12} \cdot 7^{2} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{9148592}{607257} \) | = | $2^{4} \cdot 3^{-6} \cdot 7^{-2} \cdot 17^{-1} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.80071848919275807448282276221$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.21068577551459364415962127056$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9006280039443292$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9341060098310083$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.69440500528088963971638506800$ |
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| Real period: | $\Omega$ | ≈ | $0.80298009881960826733317664244$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3455603985676761817944174485 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.345560399 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.802980 \cdot 0.694405 \cdot 24}{2^2} \\ & \approx 3.345560399\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $1$ | $I_{1}$ | split 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 409 & 4 \\ 818 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1094 & 1 \\ 167 & 0 \end{array}\right),\left(\begin{array}{rr} 953 & 4 \\ 478 & 9 \end{array}\right),\left(\begin{array}{rr} 1425 & 4 \\ 1424 & 5 \end{array}\right),\left(\begin{array}{rr} 1072 & 361 \\ 357 & 1072 \end{array}\right)$.
The torsion field $K:=\Q(E[1428])$ is a degree-$60643344384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1428\Z)$.
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 | $2$ | \( 153 = 3^{2} \cdot 17 \) |
| $3$ | additive | $6$ | \( 476 = 2^{2} \cdot 7 \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 4284.c
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1428.e2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.119952.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.66532342173696.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.7017082963632.8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | split | ss | ord | split | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 2 | 1,1 | 1 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 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.