Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-4938980x-23632928304\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-4938980xz^2-23632928304z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6400918755x-1102521889173474\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1197574767513640/303934997809, 22212561164422917074868/167560276097095127)$ | $31.116077790168383716492275605$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-233525924999722427584512$ | = | $-1 \cdot 2^{12} \cdot 3 \cdot 7^{9} \cdot 11^{8} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{528330801625}{9259880448} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-1} \cdot 5^{3} \cdot 7^{-3} \cdot 11 \cdot 13^{-3} \cdot 727^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1651074578803999992287739164$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.59355553482049631730146849270$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9476663524324863$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.698963733641945$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $31.116077790168383716492275605$ |
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| Real period: | $\Omega$ | ≈ | $0.042631091748124852730138004509$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.3060494684578335433997618838 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.306049468 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.042631 \cdot 31.116078 \cdot 4}{1^2} \\ & \approx 5.306049468\end{aligned}$$
Modular invariants
Modular form 462462.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49268736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $11$ | $1$ | $IV^{*}$ | 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 $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 458 & 639 \\ 1 & 274 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 935 & 1086 \\ 621 & 1073 \end{array}\right),\left(\begin{array}{rr} 925 & 6 \\ 591 & 19 \end{array}\right),\left(\begin{array}{rr} 1087 & 6 \\ 1086 & 7 \end{array}\right),\left(\begin{array}{rr} 1089 & 1090 \\ 1082 & 1085 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 547 & 6 \\ 549 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$15216574464$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 | $4$ | \( 231231 = 3 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $32$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $52$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | 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 $d=$
3.
Its isogeny class 462462z
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 66066ch2, its twist by $77$.
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.