Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+2375x-565316\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+2375xz^2-565316z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3078621x-26384607522\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(746, 20035)$ | $2.3032828552994787469478644113$ | $\infty$ |
Integral points
\( \left(746, 20035\right) \), \( \left(746, -20782\right) \)
Invariants
| Conductor: | $N$ | = | \( 16562 \) | = | $2 \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-139013979339872$ | = | $-1 \cdot 2^{5} \cdot 7^{11} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{5735339}{537824} \) | = | $2^{-5} \cdot 7^{-5} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3932504143107816290036742100$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.22094199958225920756237402211$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0058729515475906$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.118854996743016$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3032828552994787469478644113$ |
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| Real period: | $\Omega$ | ≈ | $0.27644161408273882495632981727$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0937858416646982012360345854 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.093785842 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.276442 \cdot 2.303283 \cdot 8}{1^2} \\ & \approx 5.093785842\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 57600 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $7$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $13$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 |
|---|---|---|
| $5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 287 & 10 \\ 65 & 93 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 911 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3119 & 3630 \\ 1035 & 3589 \end{array}\right),\left(\begin{array}{rr} 1821 & 10 \\ 1825 & 51 \end{array}\right),\left(\begin{array}{rr} 3631 & 10 \\ 3630 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3585 & 3521 \end{array}\right),\left(\begin{array}{rr} 2737 & 1830 \\ 3600 & 3401 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 637 = 7^{2} \cdot 13 \) |
| $5$ | good | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $50$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 16562u
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 2366h2, its twist by $-7$.
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 |
|---|---|---|---|
| $3$ | 3.1.728.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.0.13456625.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.4.1378893649524941167085027027130126953125.1 | \(\Z/5\Z\) | 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 | nonsplit | ord | ord | add | ord | add | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 1 | - | 3 | - | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 1 | - | 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.