Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-431933x+99139237\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-431933xz^2+99139237z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-34986600x+72167544000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-453, 14200)$ | $0.79774257443214462441853785837$ | $\infty$ |
Integral points
\((-453,\pm 14200)\)
Invariants
| Conductor: | $N$ | = | \( 454400 \) | = | $2^{8} \cdot 5^{2} \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $923765427712000000$ | = | $2^{15} \cdot 5^{6} \cdot 71^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{17406197775296}{1804229351} \) | = | $2^{6} \cdot 11^{3} \cdot 19^{3} \cdot 31^{3} \cdot 71^{-5}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1825160948607008563752614315$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.51136316294371903230334161307$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9556798643741012$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.879843247554598$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.79774257443214462441853785837$ |
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| Real period: | $\Omega$ | ≈ | $0.27127338127015155804575321192$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 20 $ = $ 2\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3281265109872685349853208909 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.328126511 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.271273 \cdot 0.797743 \cdot 20}{1^2} \\ & \approx 4.328126511\end{aligned}$$
Modular invariants
Modular form 454400.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3768320 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 8 | 15 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $71$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5Cs | 5.30.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28400 = 2^{4} \cdot 5^{2} \cdot 71 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 16 & 35 \\ 1765 & 3861 \end{array}\right),\left(\begin{array}{rr} 16831 & 50 \\ 21535 & 23121 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 28351 & 50 \\ 28350 & 51 \end{array}\right),\left(\begin{array}{rr} 21257 & 28350 \\ 2579 & 23143 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12199 & 28350 \\ 18896 & 16423 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right),\left(\begin{array}{rr} 3599 & 50 \\ 26025 & 1199 \end{array}\right)$.
The torsion field $K:=\Q(E[28400])$ is a degree-$153899827200000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28400\Z)$.
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 | $4$ | \( 1775 = 5^{2} \cdot 71 \) |
| $5$ | additive | $14$ | \( 256 = 2^{8} \) |
| $71$ | split multiplicative | $72$ | \( 6400 = 2^{8} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 454400.m
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 18176.e2, 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.