Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+5x+10\)
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(homogenize, simplify) |
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\(y^2z=x^3+5xz^2+10z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5x+10\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1, 4)$ | $0.12837506294605086906217592378$ | $\infty$ |
Integral points
\((-1,\pm 2)\), \((1,\pm 4)\), \((6,\pm 16)\), \((9,\pm 28)\)
Invariants
| Conductor: | $N$ | = | \( 400 \) | = | $2^{4} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $-51200$ | = | $-1 \cdot 2^{11} \cdot 5^{2} $ |
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| j-invariant: | $j$ | = | \( 270 \) | = | $2 \cdot 3^{3} \cdot 5$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.41352042872867736321470958332$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3171449963143106259472989169$ |
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| $abc$ quality: | $Q$ | ≈ | $1.018975235452531$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0256902266652905$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.12837506294605086906217592378$ |
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| Real period: | $\Omega$ | ≈ | $2.5292107620580267572591609418$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.2987503631321138881888327747 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.298750363 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.529211 \cdot 0.128375 \cdot 4}{1^2} \\ & \approx 1.298750363\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $5$ | $1$ | $II$ | 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 | 8.2.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 8.2.0.a.1, level \( 8 = 2^{3} \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 6 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 7 & 0 \end{array}\right),\left(\begin{array}{rr} 7 & 2 \\ 7 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[8])$ is a degree-$768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $5$ | additive | $10$ | \( 16 = 2^{4} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 400h consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 200e1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.139968000000.3 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.1310720000000000.1 | \(\Z/4\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1,3 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Additional information
Mazur and Swinnerton-Dyer reported in 1974 [https://link.springer.com/article/10.1007/BF01389997] on numerical computation suggesting that the modular parametrization $X_0(400) \to E$ of this curve has a multiple branch point at the fixed point $i/20$ of the involution $w : z \leftrightarrow 1/(400z)$. Equivalently, the associated modular form, which automatically vanishes at $i/20$ because $E$ has sign $-1$, seems to actually have a triple zero there. As far as I know, this has yet to be proved (or disproved).
This curve also admits a Belyi map of degree $5$: the degree-$5$ function $f := (x-5)y$ is ramified only above $\infty$ (a quintuple preimage at the origin of $E$) and $\pm 16$ (quadruple images at the generators $(1,\mp4)$ of the Mordell-Weil group). Since $E$ has positive rank, this gives an infinite family of "near-misses" for the ABC conjecture by evaluating the solution $(f-16, 32, f+16)$ of $A+B=C$ at rational points of $E$ and removing common factors; see Elkies 1991 [https://academic.oup.com/imrn/article-abstract/1991/7/99/667898?redirectedFrom=PDF], page 107. This Belyi map now duly appears in the LMFDB [http://beta.lmfdb.org/Belyi/5T5/%5B5%2C4%2C4%5D/5/41/41/g1/a].