Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+586367x-55345137\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+586367xz^2-55345137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+47495700x-40489092000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(338, 13475)$ | $2.0902626166977392407436967079$ | $\infty$ |
| $(93, 0)$ | $0$ | $2$ |
Integral points
\( \left(93, 0\right) \), \((338,\pm 13475)\), \((2593,\pm 137500)\)
Invariants
| Conductor: | $N$ | = | \( 431200 \) | = | $2^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-14235529000000000000$ | = | $-1 \cdot 2^{12} \cdot 5^{12} \cdot 7^{6} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2961169856}{1890625} \) | = | $2^{6} \cdot 5^{-6} \cdot 11^{-2} \cdot 359^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3641564035382376189259377808$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.10666480776641453034435037900$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9501666666067821$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.966194439238829$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0902626166977392407436967079$ |
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| Real period: | $\Omega$ | ≈ | $0.12760581146762902731921625276$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.2676745181466341115153154494 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.267674518 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.127606 \cdot 2.090263 \cdot 64}{2^2} \\ & \approx 4.267674518\end{aligned}$$
Modular invariants
Modular form 431200.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6635520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 5 | 12 | 0 |
| $5$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 | 4.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2463 & 2198 \\ 0 & 3079 \end{array}\right),\left(\begin{array}{rr} 879 & 0 \\ 0 & 3079 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1541 & 1764 \\ 882 & 449 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2521 & 448 \\ 2842 & 897 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 771 & 1764 \\ 385 & 1 \end{array}\right),\left(\begin{array}{rr} 3073 & 8 \\ 3072 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[3080])$ is a degree-$408748032000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3080\Z)$.
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 | $2$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 17248 = 2^{5} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $26$ | \( 8800 = 2^{5} \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 39200 = 2^{5} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 431200.bb
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1760.a2, its twist by $-35$.
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.