Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+677626x-380562778\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+677626xz^2-380562778z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+878203917x-17758171570482\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1751/4, -1755/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 451770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-82497936297808249350$ | = | $-1 \cdot 2 \cdot 3 \cdot 5^{2} \cdot 11^{8} \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{13411719834479}{32153832150} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-2} \cdot 11^{-8} \cdot 23^{3} \cdot 1033^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5055475968045158548702424856$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.70008864048240363268619465008$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0027749166866642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.073270641300245$ | |||
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: | $\Omega$ | ≈ | $0.099414903251632696886727399676$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.1812769040522463003752767896 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.181276904 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.099415 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 3.181276904\end{aligned}$$
Modular invariants
Modular form 451770.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13271040 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $37$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 97680 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 44879 & 0 \\ 0 & 97679 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 97676 & 97677 \end{array}\right),\left(\begin{array}{rr} 13543 & 34336 \\ 24494 & 5625 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 35521 & 34336 \\ 57128 & 79329 \end{array}\right),\left(\begin{array}{rr} 22016 & 29045 \\ 31635 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 11101 & 34336 \\ 17168 & 78885 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 97582 & 97667 \end{array}\right),\left(\begin{array}{rr} 45548 & 53465 \\ 85803 & 93722 \end{array}\right),\left(\begin{array}{rr} 97665 & 16 \\ 97664 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[97680])$ is a degree-$70934367043584000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97680\Z)$.
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 | $4$ | \( 4107 = 3 \cdot 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 150590 = 2 \cdot 5 \cdot 11 \cdot 37^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 90354 = 2 \cdot 3 \cdot 11 \cdot 37^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 41070 = 2 \cdot 3 \cdot 5 \cdot 37^{2} \) |
| $37$ | additive | $686$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 451770.t
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330.e6, its twist by $37$.
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 $0$.