Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-269863063313x-53958946691671383\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-269863063313xz^2-53958946691671383z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-349742530053675x-2517507567619029884250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(626482, 150876259)$ | $4.6248521795958673762674049137$ | $\infty$ |
| $(-1199697/4, 1199697/8)$ | $0$ | $2$ |
Integral points
\( \left(626482, 150876259\right) \), \( \left(626482, -151502741\right) \)
Invariants
| Conductor: | $N$ | = | \( 127050 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $2669190389742990960000000$ | = | $2^{10} \cdot 3^{3} \cdot 5^{7} \cdot 7^{8} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{78519570041710065450485106721}{96428056919040} \) | = | $2^{-10} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-8} \cdot 11^{-2} \cdot 2543^{3} \cdot 1683887^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.8610985843684522690275993491$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8574319917522168096962478935$ |
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| $abc$ quality: | $Q$ | ≈ | $1.052918061430929$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.7071609084836705$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6248521795958673762674049137$ |
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| Real period: | $\Omega$ | ≈ | $0.0066238988691698226717135603446$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1920 $ = $ ( 2 \cdot 5 )\cdot3\cdot2\cdot2^{3}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.704585498801274661192438654 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.704585499 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.006624 \cdot 4.624852 \cdot 1920}{2^2} \\ & \approx 14.704585499\end{aligned}$$
Modular invariants
Modular form 127050.2.a.ir
For more coefficients, see the Downloads section to the right.
| Modular degree: | 530841600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2B | 8.24.0.90 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18480 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 17018 & 4631 \\ 8107 & 12486 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18465 & 16 \\ 18464 & 17 \end{array}\right),\left(\begin{array}{rr} 15784 & 11759 \\ 8657 & 13430 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 18382 & 18467 \end{array}\right),\left(\begin{array}{rr} 6733 & 15136 \\ 14124 & 10825 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 18476 & 18477 \end{array}\right),\left(\begin{array}{rr} 3928 & 6721 \\ 1199 & 5050 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 15136 \\ 3608 & 10209 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 18479 \end{array}\right)$.
The torsion field $K:=\Q(E[18480])$ is a degree-$78479622144000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18480\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 42350 = 2 \cdot 5^{2} \cdot 7 \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 2541 = 3 \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 18150 = 2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 127050.ir
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2310.o1, its twist by $-55$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-55})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{-55})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.42693156000000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.10929447936000000.82 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.48575324160000.38 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | split | add | split | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 2 | - | 2 | - | 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 |
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.