Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-24400x+1458548\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-24400xz^2+1458548z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1976427x+1069210746\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{8}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(92, 30)$ | $1.8047053050685447270176998465$ | $\infty$ |
| $(86, 60)$ | $0$ | $8$ |
Integral points
\((-154,\pm 1260)\), \((-34,\pm 1500)\), \((46,\pm 660)\), \((86,\pm 60)\), \( \left(91, 0\right) \), \((92,\pm 30)\), \((116,\pm 450)\), \((191,\pm 1950)\), \((716,\pm 18750)\)
Invariants
| Conductor: | $N$ | = | \( 1680 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $453600000000$ | = | $2^{11} \cdot 3^{4} \cdot 5^{8} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( \frac{784478485879202}{221484375} \) | = | $2 \cdot 3^{-4} \cdot 5^{-8} \cdot 7^{-1} \cdot 71^{3} \cdot 1031^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2177491886775097582375269618$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.58236427316422655793839751713$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0076049224324828$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.644702383466793$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8047053050685447270176998465$ |
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| Real period: | $\Omega$ | ≈ | $0.91680824905271306595090925631$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2^{3}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3091374215920697318466287909 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.309137422 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.916808 \cdot 1.804705 \cdot 128}{8^2} \\ & \approx 3.309137422\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4096 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 8.48.0.159 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 560 = 2^{4} \cdot 5 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 194 & 135 \\ 395 & 294 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 462 & 547 \end{array}\right),\left(\begin{array}{rr} 545 & 16 \\ 544 & 17 \end{array}\right),\left(\begin{array}{rr} 248 & 1 \\ 559 & 10 \end{array}\right),\left(\begin{array}{rr} 436 & 421 \\ 439 & 290 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 556 & 557 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 16 \\ 456 & 129 \end{array}\right)$.
The torsion field $K:=\Q(E[560])$ is a degree-$123863040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/560\Z)$.
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 | $4$ | \( 7 \) |
| $3$ | split multiplicative | $4$ | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 1680.p
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 840.f1, its twist by $-4$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | 4.0.806400.2 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.1973822685184.2 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.24981193359360000.27 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.0.509820272640000.343 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.2.1088844664320000.12 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | split | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3,1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.