Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+1351684403x-4529382306415\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+1351684403xz^2-4529382306415z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1751782986261x-211328116237057050\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(5774, 1859303\right) \) | $4.3526356038420360034761282373$ | $\infty$ |
| \( \left(\frac{13295}{4}, -\frac{13295}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([5774:1859303:1]\) | $4.3526356038420360034761282373$ | $\infty$ |
| \([26590:-13295:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(207867, 402233040\right) \) | $4.3526356038420360034761282373$ | $\infty$ |
| \( \left(119658, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(5774, 1859303\right) \), \( \left(5774, -1865077\right) \), \( \left(568502, 429251219\right) \), \( \left(568502, -429819721\right) \)
\([5774:1859303:1]\), \([5774:-1865077:1]\), \([568502:429251219:1]\), \([568502:-429819721:1]\)
\((207867,\pm 402233040)\), \((20466075,\pm 92779661520)\)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-166917194174341682365200850944$ | = | $-1 \cdot 2^{13} \cdot 3^{10} \cdot 7^{9} \cdot 11^{6} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3820420340137317041}{2334869460099072} \) | = | $2^{-13} \cdot 3^{-10} \cdot 13^{-6} \cdot 1563281^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2964616589743516878207750855$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6380814107836814369607887389$ |
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| $abc$ quality: | $Q$ | ≈ | $1.129325119818151$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.72566887293694$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3526356038420360034761282373$ |
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| Real period: | $\Omega$ | ≈ | $0.018673557550328781600477831314$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1040 $ = $ 13\cdot( 2 \cdot 5 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $21.132589775428124999254978826 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 21.132589775 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.018674 \cdot 4.352636 \cdot 1040}{2^2} \\ & \approx 21.132589775\end{aligned}$$
Modular invariants
Modular form 462462.2.a.jy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 670924800 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $13$ | $I_{13}$ | split multiplicative | -1 | 1 | 13 | 13 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2017 & 4 \\ 1850 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1457 & 4 \\ 730 & 9 \end{array}\right),\left(\begin{array}{rr} 628 & 1 \\ 935 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1091 & 0 \end{array}\right),\left(\begin{array}{rr} 1913 & 274 \\ 272 & 1911 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$324620255232$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 847 = 7 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 11858 = 2 \cdot 7^{2} \cdot 11^{2} \) |
| $5$ | good | $2$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462462jy
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3822e2, its twist by $77$.
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.