Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-73847539x+244278350967\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-73847539xz^2+244278350967z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1181560627x+15632632901262\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(22582957/8464, 199362507719/778688)$ | $13.509426374563800988502172978$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5422698685591226$ | = | $-1 \cdot 2 \cdot 13^{4} \cdot 37^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{607782291676209}{74} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{2} \cdot 37^{-1} \cdot 5107^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8820478205476881291030754144$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.22160574507173032823453176503$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0553700072005723$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.0568601677509895$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $13.509426374563800988502172978$ |
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| Real period: | $\Omega$ | ≈ | $0.24374151912187125971751877881$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.5856162140025092513607685868 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.585616214 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.243742 \cdot 13.509426 \cdot 2}{1^2} \\ & \approx 6.585616214\end{aligned}$$
Modular invariants
Modular form 462722.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23901696 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $37$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 26936 = 2^{3} \cdot 7 \cdot 13 \cdot 37 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 6735 & 14 \\ 20209 & 99 \end{array}\right),\left(\begin{array}{rr} 24751 & 26922 \\ 11641 & 26837 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 14513 & 10 \\ 8386 & 109 \end{array}\right),\left(\begin{array}{rr} 13469 & 14 \\ 13475 & 99 \end{array}\right),\left(\begin{array}{rr} 26923 & 14 \\ 26922 & 15 \end{array}\right),\left(\begin{array}{rr} 20203 & 13482 \\ 0 & 16355 \end{array}\right)$.
The torsion field $K:=\Q(E[26936])$ is a degree-$1540404266139648$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/26936\Z)$.
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 | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $62$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $722$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 462722e
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 12506d1, its twist by $37$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.