Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-133623x+1085370678\)
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(homogenize, simplify) |
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\(y^2z=x^3-133623xz^2+1085370678z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-133623x+1085370678\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-446, 32500)$ | $2.4991815532108607184943829729$ | $\infty$ |
| $(-1071, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1071, 0\right) \), \((-446,\pm 32500)\), \((4221,\pm 275184)\)
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-508756053159900000000$ | = | $-1 \cdot 2^{8} \cdot 3^{9} \cdot 5^{8} \cdot 7^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{445090032}{858203125} \) | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{-8} \cdot 13^{-3} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6521781696617887227243599128$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.39316575825975292868042819908$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1339924153962322$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.229321271134179$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4991815532108607184943829729$ |
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| Real period: | $\Omega$ | ≈ | $0.13292354960610173370738973710$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot2\cdot2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9864009979545386915711171895 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.986400998 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.132924 \cdot 2.499182 \cdot 48}{2^2} \\ & \approx 3.986400998\end{aligned}$$
Modular invariants
Modular form 458640.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 19906560 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 \( 156 = 2^{2} \cdot 3 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 134 & 1 \\ 11 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 153 & 4 \\ 152 & 5 \end{array}\right),\left(\begin{array}{rr} 56 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 121 & 40 \\ 116 & 39 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[156])$ is a degree-$10063872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/156\Z)$.
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 | $2$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $2$ | \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 458640.v
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4680.h2, its twist by $-84$.
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.