Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+689263x-674504745\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+689263xz^2-674504745z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+893284821x-31483092646746\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(82215373/2401, 743681014924/117649)$ | $13.930660586002591500522972353$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462722 \) | = | $2 \cdot 13^{2} \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-217665952764240332456$ | = | $-1 \cdot 2^{3} \cdot 13^{9} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1331}{8} \) | = | $2^{-3} \cdot 11^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5843555374631993450114072426$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1448154369550654292127561741$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9357683713670925$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.153581199067143$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $13.930660586002591500522972353$ |
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| Real period: | $\Omega$ | ≈ | $0.088714188075927938564668068946$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 3\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4150834594913024720230682910 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.415083459 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.088714 \cdot 13.930661 \cdot 6}{1^2} \\ & \approx 7.415083459\end{aligned}$$
Modular invariants
Modular form 462722.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16117920 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $37$ | $1$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3Ns | 3.6.0.1 |
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 57720 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 37 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 19981 & 49950 \\ 12210 & 10657 \end{array}\right),\left(\begin{array}{rr} 24976 & 47175 \\ 3885 & 37186 \end{array}\right),\left(\begin{array}{rr} 9693 & 25123 \\ 21830 & 33447 \end{array}\right),\left(\begin{array}{rr} 14431 & 18870 \\ 10545 & 52171 \end{array}\right),\left(\begin{array}{rr} 8881 & 0 \\ 0 & 53281 \end{array}\right),\left(\begin{array}{rr} 48841 & 48840 \\ 8880 & 48841 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 49950 & 1 \end{array}\right),\left(\begin{array}{rr} 28861 & 0 \\ 15540 & 28861 \end{array}\right),\left(\begin{array}{rr} 11656 & 33855 \\ 6105 & 18871 \end{array}\right),\left(\begin{array}{rr} 54391 & 9990 \\ 0 & 54391 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 48360 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 15540 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8880 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 23088 \\ 53280 & 1 \end{array}\right),\left(\begin{array}{rr} 9361 & 48360 \\ 9360 & 9361 \end{array}\right),\left(\begin{array}{rr} 9359 & 0 \\ 0 & 57719 \end{array}\right),\left(\begin{array}{rr} 19241 & 18870 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19241 & 0 \\ 0 & 19241 \end{array}\right),\left(\begin{array}{rr} 22201 & 38850 \\ 36630 & 11101 \end{array}\right)$.
The torsion field $K:=\Q(E[57720])$ is a degree-$2934103364075520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/57720\Z)$.
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 | $4$ | \( 17797 = 13 \cdot 37^{2} \) |
| $3$ | good | $2$ | \( 231361 = 13^{2} \cdot 37^{2} \) |
| $13$ | additive | $62$ | \( 2738 = 2 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 338 = 2 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 462722.k
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 338.d2, its twist by $481$.
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.