Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-14075x+1042250\)
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(homogenize, simplify) |
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\(y^2z=x^3-14075xz^2+1042250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-14075x+1042250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(295, 4750)$ | $0.74862554535957849949296224572$ | $\infty$ |
| $(10, 950)$ | $1.4406980922678248520505169771$ | $\infty$ |
Integral points
\((10,\pm 950)\), \((86,\pm 684)\), \((95,\pm 750)\), \((295,\pm 4750)\)
Invariants
| Conductor: | $N$ | = | \( 402800 \) | = | $2^{4} \cdot 5^{2} \cdot 19 \cdot 53$ |
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| Discriminant: | $\Delta$ | = | $-290821600000000$ | = | $-1 \cdot 2^{11} \cdot 5^{8} \cdot 19^{3} \cdot 53 $ |
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| j-invariant: | $j$ | = | \( -\frac{9636491538}{9088175} \) | = | $-1 \cdot 2 \cdot 3^{3} \cdot 5^{-2} \cdot 19^{-3} \cdot 53^{-1} \cdot 563^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4716898245047562939409453374$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.031585952774422906341436226117$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7991590159763713$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1951267529889824$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0552988035300554405423841076$ |
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| Real period: | $\Omega$ | ≈ | $0.49940792430231680529459916425$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $12.648590039751922129934321016 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.648590040 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.499408 \cdot 1.055299 \cdot 24}{1^2} \\ & \approx 12.648590040\end{aligned}$$
Modular invariants
Modular form 402800.2.a.l
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $53$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8056 = 2^{3} \cdot 19 \cdot 53 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4029 & 2 \\ 4029 & 3 \end{array}\right),\left(\begin{array}{rr} 6785 & 2 \\ 6785 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8055 & 0 \end{array}\right),\left(\begin{array}{rr} 2281 & 2 \\ 2281 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8055 & 2 \\ 8054 & 3 \end{array}\right),\left(\begin{array}{rr} 2015 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8056])$ is a degree-$731755749703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8056\Z)$.
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 | $4$ | \( 25175 = 5^{2} \cdot 19 \cdot 53 \) |
| $3$ | good | $2$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $5$ | additive | $18$ | \( 16112 = 2^{4} \cdot 19 \cdot 53 \) |
| $19$ | split multiplicative | $20$ | \( 21200 = 2^{4} \cdot 5^{2} \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 402800l consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 40280b1, its twist by $-20$.
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.