Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-4435x-56657\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-4435xz^2-56657z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5748435x-2557165842\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-29, 235)$ | $1.1773787380049257055945422024$ | $\infty$ |
| $(97, 613)$ | $1.8320542831866886431729087687$ | $\infty$ |
| $(-53/4, 53/8)$ | $0$ | $2$ |
Integral points
\( \left(-57, 151\right) \), \( \left(-57, -94\right) \), \( \left(-49, 239\right) \), \( \left(-49, -190\right) \), \( \left(-29, 235\right) \), \( \left(-29, -206\right) \), \( \left(77, 218\right) \), \( \left(77, -295\right) \), \( \left(97, 613\right) \), \( \left(97, -710\right) \), \( \left(237, 3385\right) \), \( \left(237, -3622\right) \), \( \left(10110337, 32142538810\right) \), \( \left(10110337, -32152649147\right) \)
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $4287143442582$ | = | $2 \cdot 3^{4} \cdot 7^{6} \cdot 11^{3} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{61629875}{27378} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{3} \cdot 13^{-2} \cdot 79^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1194906431674202273239768903$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.45293824955982906124418537591$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9071218849981642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.8215962915228294$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0644704026915028265974896428$ |
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| Real period: | $\Omega$ | ≈ | $0.60960934526357592867506386483$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.068163604006383830394721311 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.068163604 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.609609 \cdot 2.064470 \cdot 32}{2^2} \\ & \approx 10.068163604\end{aligned}$$
Modular invariants
Modular form 462462.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1144 = 2^{3} \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 316 & 1 \\ 207 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 353 & 4 \\ 706 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 571 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1141 & 4 \\ 1140 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 716 & 433 \\ 1001 & 144 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[1144])$ is a degree-$44281036800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1144\Z)$.
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 | $4$ | \( 539 = 7^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $42$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35574 = 2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 462462.y
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 9438.n1, its twist by $-7$.
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.