Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-11600212x+15199218761\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-11600212xz^2+15199218761z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15033874779x+709360258643190\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(161845/81, 543043/729)$ | $7.9542070463681986570343098766$ | $\infty$ |
| $(7955/4, -7959/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 462462 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $40318168107263682636$ | = | $2^{2} \cdot 3^{12} \cdot 7^{7} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{828279937799497}{193444524} \) | = | $2^{-2} \cdot 3^{-12} \cdot 7^{-1} \cdot 13^{-1} \cdot 93913^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7525460437385978736048072171$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.58064333281175594902115905640$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9834700067632124$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.6313802758552$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.9542070463681986570343098766$ |
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| Real period: | $\Omega$ | ≈ | $0.19881068230766307763918937503$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.651050640839064145451532388 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.651050641 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.198811 \cdot 7.954207 \cdot 32}{2^2} \\ & \approx 12.651050641\end{aligned}$$
Modular invariants
Modular form 462462.2.a.hm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23592960 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 23396 & 4367 \\ 21505 & 21834 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 24018 & 24019 \end{array}\right),\left(\begin{array}{rr} 8009 & 13112 \\ 14564 & 4401 \end{array}\right),\left(\begin{array}{rr} 14620 & 19657 \\ 4895 & 2190 \end{array}\right),\left(\begin{array}{rr} 19655 & 0 \\ 0 & 24023 \end{array}\right),\left(\begin{array}{rr} 12288 & 22121 \\ 18865 & 10176 \end{array}\right),\left(\begin{array}{rr} 24017 & 8 \\ 24016 & 9 \end{array}\right),\left(\begin{array}{rr} 1915 & 1914 \\ 12298 & 17755 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$1071246842265600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
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 | $4$ | \( 77077 = 7^{2} \cdot 11^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 154154 = 2 \cdot 7^{2} \cdot 11^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $13$ | split 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 and 4.
Its isogeny class 462462.hm
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 546.b1, 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.