Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-161697x+24959301\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-161697xz^2+24959301z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-209559987x+1167644543886\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(230, -69)$ | $2.0053357000304478165689342671$ | $\infty$ |
Integral points
\( \left(230, -69\right) \), \( \left(230, -161\right) \)
Invariants
| Conductor: | $N$ | = | \( 400710 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $4388575920$ | = | $2^{4} \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{1295169456680034481}{12156720} \) | = | $2^{-4} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{3} \cdot 13^{3} \cdot 19 \cdot 37^{-3} \cdot 67^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4300196367809368716641398083$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.93927980691986346166263523632$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0551675433968963$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6891772781649173$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0053357000304478165689342671$ |
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| Real period: | $\Omega$ | ≈ | $0.96213719046028542888368778868$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8588161127140095588391096273 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.858816113 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.962137 \cdot 2.005336 \cdot 2}{1^2} \\ & \approx 3.858816113\end{aligned}$$
Modular invariants
Modular form 400710.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1586304 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $37$ | $1$ | $I_{3}$ | 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 \( 42180 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \cdot 37 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 39963 & 2 \\ 6670 & 7 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 3518 & 38667 \\ 1 & 10546 \end{array}\right),\left(\begin{array}{rr} 21091 & 6 \\ 21093 & 19 \end{array}\right),\left(\begin{array}{rr} 23941 & 6 \\ 29643 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 42175 & 6 \\ 42174 & 7 \end{array}\right),\left(\begin{array}{rr} 8437 & 6 \\ 25311 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[42180])$ is a degree-$31013633772748800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/42180\Z)$.
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 | $4$ | \( 200355 = 3 \cdot 5 \cdot 19^{2} \cdot 37 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 80142 = 2 \cdot 3 \cdot 19^{2} \cdot 37 \) |
| $19$ | additive | $74$ | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 10830 = 2 \cdot 3 \cdot 5 \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 400710l
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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.