Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-408x-3281\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-408xz^2-3281z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-529200x-146718000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-55162259/4588164, 50009504273/9827847288)$ | $17.435048307668307308577546650$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 136525 \) | = | $5^{2} \cdot 43 \cdot 127$ |
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| Discriminant: | $\Delta$ | = | $85328125$ | = | $5^{6} \cdot 43 \cdot 127 $ |
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| j-invariant: | $j$ | = | \( \frac{481890304}{5461} \) | = | $2^{12} \cdot 7^{6} \cdot 43^{-1} \cdot 127^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.33529777634326145855179628805$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.46942117987378872874858337856$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7819297671826626$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.507543554122664$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $17.435048307668307308577546650$ |
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| Real period: | $\Omega$ | ≈ | $1.0627797845456695654958022140$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $18.529616883967064419499353342 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 18.529616884 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.062780 \cdot 17.435048 \cdot 1}{1^2} \\ & \approx 18.529616884\end{aligned}$$
Modular invariants
Modular form 136525.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 52080 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $43$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $127$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10922 = 2 \cdot 43 \cdot 127 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5335 & 2 \\ 5335 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10921 & 0 \end{array}\right),\left(\begin{array}{rr} 10921 & 2 \\ 10920 & 3 \end{array}\right),\left(\begin{array}{rr} 5591 & 2 \\ 5591 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[10922])$ is a degree-$2584019272310784$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10922\Z)$.
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 |
|---|---|---|---|
| $5$ | additive | $14$ | \( 5461 = 43 \cdot 127 \) |
| $43$ | split multiplicative | $44$ | \( 3175 = 5^{2} \cdot 127 \) |
| $127$ | split multiplicative | $128$ | \( 1075 = 5^{2} \cdot 43 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 136525.c consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 5461.a1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.21844.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.2605772594896.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 127 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | add | ord | ss | ord | ord | ss | ord | ord | ord | ord | ord | split | ord | split |
| $\lambda$-invariant(s) | 2,1 | 9 | - | 3 | 1,1 | 1 | 5 | 1,1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 |
| $\mu$-invariant(s) | 0,0 | 0 | - | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.