Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-4820x-318791\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-4820xz^2-318791z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6246747x-14779803402\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-36184868728996$ | = | $-1 \cdot 2^{2} \cdot 13^{6} \cdot 37^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{1369}{4} \) | = | $-1 \cdot 2^{-2} \cdot 37^{2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2885924342411951733709514302$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1975215487043146761118241809$ |
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| $abc$ quality: | $Q$ | ≈ | $0.910040393716602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.978569498756785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.26512105286875534066991892342$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.1209684229500427253593513874 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.120968423 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.265121 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 2.120968423\end{aligned}$$
Modular invariants
Modular form 462722.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1105920 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $37$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 52.4.0-4.a.1.1, level \( 52 = 2^{2} \cdot 13 \), index $4$, genus $0$, and generators
$\left(\begin{array}{rr} 11 & 0 \\ 0 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 49 & 4 \\ 48 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 47 & 45 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 13 \\ 0 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[52])$ is a degree-$628992$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/52\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $86$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $470$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462722q consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2738a1, its twist by $13$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.