Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-50319363x+137388609538\)
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(homogenize, simplify) |
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\(y^2z=x^3-50319363xz^2+137388609538z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-50319363x+137388609538\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5186, 126126)$ | $4.3090094331933155663948458889$ | $\infty$ |
| $(4094, 0)$ | $0$ | $2$ |
| $(4097, 0)$ | $0$ | $2$ |
Integral points
\( \left(-8191, 0\right) \), \( \left(4094, 0\right) \), \( \left(4097, 0\right) \), \((5186,\pm 126126)\)
Invariants
| Conductor: | $N$ | = | \( 65520 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $3281521236089241600$ | = | $2^{28} \cdot 3^{10} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4718909406724749250561}{1098974822400} \) | = | $2^{-16} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 433^{3} \cdot 38737^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9327308462845184046346543156$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6902775213905182495197995757$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0081936394487907$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.844423578055518$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3090094331933155663948458889$ |
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| Real period: | $\Omega$ | ≈ | $0.20010436734045292705303540394$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.8980128519455365941959392477 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.898012852 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.200104 \cdot 4.309009 \cdot 128}{4^2} \\ & \approx 6.898012852\end{aligned}$$
Modular invariants
Modular form 65520.2.a.bx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3145728 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{20}^{*}$ | additive | -1 | 4 | 28 | 16 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 16.48.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 17365 & 14568 \\ 19518 & 21709 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 4381 & 14568 \\ 14490 & 7237 \end{array}\right),\left(\begin{array}{rr} 7265 & 14544 \\ 7056 & 671 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 18721 & 7296 \\ 3126 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 7285 & 7296 \\ 7284 & 5473 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 21839 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 13104 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 65520de
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730v2, its twist by $12$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-455})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{455})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.888731494560000.189 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.49787136.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.1563886116000000.5 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | nonsplit | split | ord | split | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 1 | 2 | 1 | 2 | 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 | 0,0 |
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.