Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-539109x+152510121\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-539109xz^2+152510121z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8625747x+9752021998\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-564, 16923\right) \) | $0.14412854756240210112830954815$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-564:16923:1]\) | $0.14412854756240210112830954815$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2257, 133128\right) \) | $0.14412854756240210112830954815$ | $\infty$ |
Integral points
\( \left(-564, 16923\right) \), \( \left(-564, -16359\right) \), \( \left(210, 6861\right) \), \( \left(210, -7071\right) \), \( \left(511, 2948\right) \), \( \left(511, -3459\right) \), \( \left(1285, 39111\right) \), \( \left(1285, -40396\right) \)
\([-564:16923:1]\), \([-564:-16359:1]\), \([210:6861:1]\), \([210:-7071:1]\), \([511:2948:1]\), \([511:-3459:1]\), \([1285:39111:1]\), \([1285:-40396:1]\)
\((-2257,\pm 133128)\), \((839,\pm 55728)\), \((2043,\pm 25628)\), \((5139,\pm 318028)\)
Invariants
| Conductor: | $N$ | = | \( 774 \) | = | $2 \cdot 3^{2} \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2377869209964036$ | = | $-1 \cdot 2^{2} \cdot 3^{7} \cdot 43^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{23769846831649063249}{3261823333284} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 13^{3} \cdot 43^{-7} \cdot 221173^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9682281419925942171459366292$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4189219976585393714483140107$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0440647512200927$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.698460050313475$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.14412854756240210112830954815$ |
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| Real period: | $\Omega$ | ≈ | $0.44310020709000863212531192490$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 28 $ = $ 2\cdot2\cdot7 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7881748996295108356176826993 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.788174900 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.443100 \cdot 0.144129 \cdot 28}{1^2} \\ & \approx 1.788174900\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9408 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $43$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
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 |
|---|---|---|---|
| $7$ | 7B.6.3 | 7.24.0.2 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3612 = 2^{2} \cdot 3 \cdot 7 \cdot 43 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3599 & 14 \\ 3598 & 15 \end{array}\right),\left(\begin{array}{rr} 2948 & 7 \\ 581 & 3606 \end{array}\right),\left(\begin{array}{rr} 1196 & 3605 \\ 1211 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 1799 & 3606 \end{array}\right),\left(\begin{array}{rr} 1809 & 3104 \\ 3598 & 2801 \end{array}\right)$.
The torsion field $K:=\Q(E[3612])$ is a degree-$322962038784$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3612\Z)$.
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 | $4$ | \( 387 = 3^{2} \cdot 43 \) |
| $3$ | additive | $8$ | \( 86 = 2 \cdot 43 \) |
| $7$ | good | $2$ | \( 18 = 2 \cdot 3^{2} \) |
| $43$ | split multiplicative | $44$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 774d
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 258f2, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.516.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.137388096.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | \(\Q(\zeta_{21})^+\) | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.9690085451952.4 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $14$ | 14.0.4429032014536033185792.3 | \(\Z/7\Z\) | not in database |
| $18$ | 18.6.2419538706625713301455840792576.1 | \(\Z/14\Z\) | 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 | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | split | ord |
| $\lambda$-invariant(s) | 3 | - | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 | 2 | 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.