Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-7521400x-7936748048\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-7521400xz^2-7936748048z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-609233427x-5787717027246\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1591, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1591, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 29040 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $2057845257600000000$ | = | $2^{13} \cdot 3 \cdot 5^{8} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{6484907238722641}{283593750} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-8} \cdot 11^{-2} \cdot 186481^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5921834574415342141343985606$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.70008864048240363268619465016$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0074430799323078$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.752327877185946$ | |||
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: | $\Omega$ | ≈ | $0.091164555335102327486012982962$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot1\cdot2^{3}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.4586328853616372397762077274 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.458632885 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.091165 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 1.458632885\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 737280 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
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 | 8.24.0.93 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1642 & 1975 \\ 987 & 1318 \end{array}\right),\left(\begin{array}{rr} 1057 & 16 \\ 536 & 129 \end{array}\right),\left(\begin{array}{rr} 227 & 2624 \\ 1216 & 315 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2542 & 2627 \end{array}\right),\left(\begin{array}{rr} 896 & 5 \\ 2595 & 2626 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 668 & 665 \\ 1323 & 1322 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2636 & 2637 \end{array}\right),\left(\begin{array}{rr} 2625 & 16 \\ 2624 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2640])$ is a degree-$38928384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2640\Z)$.
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 | $4$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 29040co
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330b5, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-66}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{11})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.179068074983424.20 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.179068074983424.36 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.77720518656.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 11 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | split | add |
| $\lambda$-invariant(s) | - | 0 | 1 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.