Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-88x-2583\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-88xz^2-2583z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-114075x-120170250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 194\right) \) | $0.81461696910621217500101772285$ | $\infty$ |
| \( \left(\frac{63}{4}, -\frac{63}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37:194:1]\) | $0.81461696910621217500101772285$ | $\infty$ |
| \([126:-63:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1335, 45900\right) \) | $0.81461696910621217500101772285$ | $\infty$ |
| \( \left(570, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(16, 5\right) \), \( \left(16, -21\right) \), \( \left(37, 194\right) \), \( \left(37, -231\right) \), \( \left(47, 289\right) \), \( \left(47, -336\right) \), \( \left(2672, 136789\right) \), \( \left(2672, -139461\right) \)
\([16:5:1]\), \([16:-21:1]\), \([37:194:1]\), \([37:-231:1]\), \([47:289:1]\), \([47:-336:1]\), \([2672:136789:1]\), \([2672:-139461:1]\)
\((579,\pm 2808)\), \((1335,\pm 45900)\), \((1695,\pm 67500)\), \((96195,\pm 29835000)\)
Invariants
| Conductor: | $N$ | = | \( 425 \) | = | $5^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2822265625$ | = | $-1 \cdot 5^{10} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{4826809}{180625} \) | = | $-1 \cdot 5^{-4} \cdot 13^{6} \cdot 17^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.49360503590899958494642543240$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31111392030805060235395423421$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0531441162406259$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.82988124952053$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.81461696910621217500101772285$ |
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| Real period: | $\Omega$ | ≈ | $0.62506696233567758421549733434$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.0183803086926331116116444452 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.018380309 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.625067 \cdot 0.814617 \cdot 8}{2^2} \\ & \approx 1.018380309\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.5 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 680 = 2^{3} \cdot 5 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 511 & 8 \\ 425 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 341 & 8 \\ 2 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 673 & 8 \\ 672 & 9 \end{array}\right),\left(\begin{array}{rr} 543 & 676 \\ 0 & 679 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[680])$ is a degree-$1203240960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/680\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 25 = 5^{2} \) |
| $5$ | additive | $18$ | \( 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 425d
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 85a2, its twist by $5$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.28900.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.13363360000.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.11837440000.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.2854069171875.2 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | ord | ord | add | ord | ord | ord | nonsplit | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 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.