Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-343035x-77264550\)
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(homogenize, simplify) |
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\(y^2z=x^3-343035xz^2-77264550z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-343035x-77264550\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-341, 242\right) \) | $1.7649229874563859147011793520$ | $\infty$ |
| \( \left(-330, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-341:242:1]\) | $1.7649229874563859147011793520$ | $\infty$ |
| \([-330:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-341, 242\right) \) | $1.7649229874563859147011793520$ | $\infty$ |
| \( \left(-330, 0\right) \) | $0$ | $2$ |
Integral points
\((-341,\pm 242)\), \( \left(-330, 0\right) \), \((826,\pm 14246)\), \((1155,\pm 32670)\)
\([-341:\pm 242:1]\), \([-330:0:1]\), \([826:\pm 14246:1]\), \([1155:\pm 32670:1]\)
\((-341,\pm 242)\), \( \left(-330, 0\right) \), \((826,\pm 14246)\), \((1155,\pm 32670)\)
Invariants
| Conductor: | $N$ | = | \( 435600 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $4463313300864000$ | = | $2^{10} \cdot 3^{9} \cdot 5^{3} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( 1000188 \) | = | $2^{2} \cdot 3^{6} \cdot 7^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9223228114466230266008163258$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0805661700287906988078059921$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0027732557726232$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8392271068835417$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7649229874563859147011793520$ |
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| Real period: | $\Omega$ | ≈ | $0.19728361422100917769413817816$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7855230862970932702572959531 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.785523086 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.197284 \cdot 1.764923 \cdot 32}{2^2} \\ & \approx 2.785523086\end{aligned}$$
Modular invariants
Modular form 435600.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3932160 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.30 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 88 & 3 \\ 37 & 104 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 102 & 19 \\ 41 & 98 \end{array}\right),\left(\begin{array}{rr} 65 & 92 \\ 112 & 69 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 113 & 8 \\ 112 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 112 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 36 & 109 \\ 53 & 68 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $2$ | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| $3$ | additive | $2$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $10$ | \( 17424 = 2^{4} \cdot 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 435600bc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1800e2, its twist by $44$.
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.