Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-761193x+255548548\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-761193xz^2+255548548z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-986505507x+11925832583646\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-478, 22827)$ | $1.7485614446091709933575379927$ | $\infty$ |
Integral points
\( \left(-478, 22827\right) \), \( \left(-478, -22350\right) \)
Invariants
| Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $1127790700340040$ | = | $2^{3} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 37^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{19010647320769}{439560} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-1} \cdot 13^{3} \cdot 37^{-1} \cdot 2053^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0010694351302731847282674358$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.19561047880816096254421960029$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8763546497067568$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.012120017443941$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7485614446091709933575379927$ |
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| Real period: | $\Omega$ | ≈ | $0.45230004930230086867063455871$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot3\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.4904931312579656372919018416 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.490493131 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.452300 \cdot 1.748561 \cdot 12}{1^2} \\ & \approx 9.490493131\end{aligned}$$
Modular invariants
Modular form 451770.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7879680 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $4$ | $I_{1}^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 19537 & 6 \\ 9771 & 19 \end{array}\right),\left(\begin{array}{rr} 24421 & 6 \\ 24423 & 19 \end{array}\right),\left(\begin{array}{rr} 44879 & 48834 \\ 36957 & 48821 \end{array}\right),\left(\begin{array}{rr} 36631 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 35521 & 6 \\ 8883 & 19 \end{array}\right),\left(\begin{array}{rr} 48835 & 6 \\ 48834 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 46806 & 2041 \\ 32563 & 34614 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$53200775282688000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
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 | $4$ | \( 225885 = 3 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 75295 = 5 \cdot 11 \cdot 37^{2} \) |
| $5$ | split multiplicative | $6$ | \( 90354 = 2 \cdot 3 \cdot 11 \cdot 37^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
| $37$ | additive | $722$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 451770.y
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 12210.w2, 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.