Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-115500x+15106000\)
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(homogenize, simplify) |
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\(y^2z=x^3-115500xz^2+15106000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-115500x+15106000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(240, 1100)$ | $2.5215329434360831807343494055$ | $\infty$ |
Integral points
\((240,\pm 1100)\)
Invariants
| Conductor: | $N$ | = | \( 6350400 \) | = | $2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $32514048000000$ | = | $2^{19} \cdot 3^{4} \cdot 5^{6} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{10481625}{2} \) | = | $2^{-1} \cdot 3^{2} \cdot 5^{3} \cdot 7 \cdot 11^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5935957575585459938330390615$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.94136642389701093890916265685$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9483404325143082$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.9739910695966785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5215329434360831807343494055$ |
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| Real period: | $\Omega$ | ≈ | $0.63770712931404097112054895062$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.863996278635269235683310879 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.863996279 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.637707 \cdot 2.521533 \cdot 8}{1^2} \\ & \approx 12.863996279\end{aligned}$$
Modular invariants
Modular form 6350400.2.a.bgu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23887872 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.2 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 246 & 265 \\ 385 & 771 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 419 & 330 \\ 585 & 149 \end{array}\right),\left(\begin{array}{rr} 835 & 6 \\ 834 & 7 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 51 & 170 \\ 730 & 511 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 209 & 330 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$4459069440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 99225 = 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $8$ | \( 78400 = 2^{6} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 254016 = 2^{6} \cdot 3^{4} \cdot 7^{2} \) |
| $7$ | additive | $14$ | \( 129600 = 2^{6} \cdot 3^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 6350400.bgu
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 7938.h2, its twist by $-40$.
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.