Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-277x+1555\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-277xz^2+1555z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-359667x+77941710\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-11, 64\right) \) | $0.77505344866638871730383399017$ | $\infty$ |
| \( \left(7, 1\right) \) | $1.2417706912860407721967089201$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-11:64:1]\) | $0.77505344866638871730383399017$ | $\infty$ |
| \([7:1:1]\) | $1.2417706912860407721967089201$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-381, 12636\right) \) | $0.77505344866638871730383399017$ | $\infty$ |
| \( \left(267, 972\right) \) | $1.2417706912860407721967089201$ | $\infty$ |
Integral points
\( \left(-11, 64\right) \), \( \left(-11, -53\right) \), \( \left(7, 1\right) \), \( \left(7, -8\right) \), \( \left(15, 25\right) \), \( \left(15, -40\right) \), \( \left(21, 64\right) \), \( \left(21, -85\right) \), \( \left(141, 1600\right) \), \( \left(141, -1741\right) \)
\([-11:64:1]\), \([-11:-53:1]\), \([7:1:1]\), \([7:-8:1]\), \([15:25:1]\), \([15:-40:1]\), \([21:64:1]\), \([21:-85:1]\), \([141:1600:1]\), \([141:-1741:1]\)
\((-381,\pm 12636)\), \((267,\pm 972)\), \((555,\pm 7020)\), \((771,\pm 16092)\), \((5091,\pm 360828)\)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $162324162$ | = | $2 \cdot 3^{4} \cdot 7^{2} \cdot 11^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{398684209}{27378} \) | = | $2^{-1} \cdot 3^{-4} \cdot 7 \cdot 11 \cdot 13^{-2} \cdot 173^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.32365958382241648438668652518$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.40030798648653082380786286172$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8358632517362515$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.1841914158017066$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.88633724097134894979741283149$ |
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| Real period: | $\Omega$ | ≈ | $1.7823151228527261101440133354$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.3189290741231844389280924794 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.318929074 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.782315 \cdot 0.886337 \cdot 4}{1^2} \\ & \approx 6.318929074\end{aligned}$$
Modular invariants
Modular form 462462.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 331776 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $11$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $13$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.2 | $2$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.b.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $14$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $32$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462462c consists of this curve only.
Twists
This elliptic curve is its own minimal quadratic twist.
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.