Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-368x-35028\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-368xz^2-35028z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-477603x-1627105698\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(36, -18)$ | $0$ | $2$ |
Integral points
\( \left(36, -18\right) \)
Invariants
| Conductor: | $N$ | = | \( 400710 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-522209279100$ | = | $-1 \cdot 2^{2} \cdot 3 \cdot 5^{2} \cdot 19^{6} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( -\frac{117649}{11100} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{6} \cdot 37^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.92801905157604349222519605735$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.54420043800717673777931765859$ |
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| $abc$ quality: | $Q$ | ≈ | $0.967124143790104$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.6697300346160624$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.41002054602100664018544417371$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.82004109204201328037088834743 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.820041092 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.410021 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 0.820041092\end{aligned}$$
Modular invariants
Modular form 400710.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 570240 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $37$ | $1$ | $I_{1}$ | nonsplit 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 \( 4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1777 & 4 \\ 3554 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2777 & 1666 \\ 1664 & 2775 \end{array}\right),\left(\begin{array}{rr} 2962 & 1 \\ 2959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 4437 & 4 \\ 4436 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 3482 & 1 \\ 479 & 0 \end{array}\right),\left(\begin{array}{rr} 2221 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[4440])$ is a degree-$5373815685120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4440\Z)$.
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 | $4$ | \( 40071 = 3 \cdot 19^{2} \cdot 37 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 133570 = 2 \cdot 5 \cdot 19^{2} \cdot 37 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 80142 = 2 \cdot 3 \cdot 19^{2} \cdot 37 \) |
| $19$ | additive | $182$ | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 400710g
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1110o1, its twist by $-19$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.