Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-4083725840x+81482167512084\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-4083725840xz^2+81482167512084z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-330781793067x+59401492461688410\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-72197, 0)$ | $0$ | $2$ |
Integral points
\( \left(-72197, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 40656 \) | = | $2^{4} \cdot 3 \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $1490334758955046902617711050752$ | = | $2^{23} \cdot 3^{22} \cdot 7^{4} \cdot 11^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{779828911477214942771}{154308452600236032} \) | = | $2^{-11} \cdot 3^{-22} \cdot 7^{-4} \cdot 9204491^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.5061239162830803649272636947$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0145552811243571474635738898$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0734168110525577$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.349962527806726$ | |||
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: | $\Omega$ | ≈ | $0.025468024937563845414637091567$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 352 $ = $ 2\cdot( 2 \cdot 11 )\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.2411861945056183964880640579 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.241186195 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.025468 \cdot 1.000000 \cdot 352}{2^2} \\ & \approx 2.241186195\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 98131968 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{15}^{*}$ | additive | -1 | 4 | 23 | 11 |
| $3$ | $22$ | $I_{22}$ | split multiplicative | -1 | 1 | 22 | 22 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
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 \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 52 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 261 & 4 \\ 260 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 131 & 0 \end{array}\right),\left(\begin{array}{rr} 100 & 169 \\ 33 & 232 \end{array}\right),\left(\begin{array}{rr} 89 & 4 \\ 178 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$81100800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 11 \) |
| $3$ | split multiplicative | $4$ | \( 13552 = 2^{4} \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $42$ | \( 112 = 2^{4} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 40656da
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 5082g2, its twist by $44$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.95832.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.587761422336.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.267496753987584.19 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | add | split | ord | split | add |
| $\lambda$-invariant(s) | - | 3 | 2 | 1 | - |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 13$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.