Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-2395439x-1572797211\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-2395439xz^2-1572797211z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3104489619x-73333859335506\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1865, 20303\right) \) | $3.6339219954281625850688121540$ | $\infty$ |
| \( \left(\frac{11695}{4}, \frac{1013659}{8}\right) \) | $6.4413093627030628990759555818$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1865:20303:1]\) | $3.6339219954281625850688121540$ | $\infty$ |
| \([23390:1013659:8]\) | $6.4413093627030628990759555818$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(67155, 4586868\right) \) | $3.6339219954281625850688121540$ | $\infty$ |
| \( \left(105270, 27684558\right) \) | $6.4413093627030628990759555818$ | $\infty$ |
Integral points
\( \left(1865, 20303\right) \), \( \left(1865, -22168\right) \), \( \left(5411, 376676\right) \), \( \left(5411, -382087\right) \)
\([1865:20303:1]\), \([1865:-22168:1]\), \([5411:376676:1]\), \([5411:-382087:1]\)
\((67155,\pm 4586868)\), \((194811,\pm 81946404)\)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-187576604242057046016$ | = | $-1 \cdot 2^{11} \cdot 3^{4} \cdot 7^{4} \cdot 11^{8} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{2953559349433}{364455936} \) | = | $-1 \cdot 2^{-11} \cdot 3^{-4} \cdot 7^{2} \cdot 11 \cdot 13^{-3} \cdot 41^{3} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6246787078900328091666427123$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.37744514300601467809022941251$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9437908048663723$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.283406142396315$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $21.940624120143524397705358316$ |
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| Real period: | $\Omega$ | ≈ | $0.060260549423139267543086486676$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot2\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.9329243849985821518179592982 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.932924385 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.060261 \cdot 21.940624 \cdot 6}{1^2} \\ & \approx 7.932924385\end{aligned}$$
Modular invariants
Modular form 462462.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23417856 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 \( 104 = 2^{3} \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 53 & 2 \\ 53 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 2 \\ 102 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 79 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[104])$ is a degree-$20127744$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/104\Z)$.
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 | $4$ | \( 77077 = 7^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 11858 = 2 \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $20$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $52$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462462g consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 462462fp1, its twist by $-11$.
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.