Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-3290247005x-72641691675003\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-3290247005xz^2-72641691675003z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-52643952075x-4649120911152250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(285773139/2401, 4105417923020/117649)$ | $15.665541293578642151286564412$ | $\infty$ |
| $(-132469/4, 132465/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 76050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $557500964633437500000$ | = | $2^{5} \cdot 3^{7} \cdot 5^{10} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{71647584155243142409}{10140000} \) | = | $2^{-5} \cdot 3^{-1} \cdot 5^{-4} \cdot 11^{3} \cdot 13^{-2} \cdot 491^{3} \cdot 769^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8337632562527205076174449255$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1972634769708471065926989196$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0375259015845228$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.8827548296612955$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $15.665541293578642151286564412$ |
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| Real period: | $\Omega$ | ≈ | $0.019933907060097478563622273908$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 160 $ = $ 5\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.491017767692635276336331968 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.491017768 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.019934 \cdot 15.665541 \cdot 160}{2^2} \\ & \approx 12.491017768\end{aligned}$$
Modular invariants
Modular form 76050.2.a.gb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41287680 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $13$ | $2$ | $I_{2}^{*}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 272 & 117 \\ 1235 & 1078 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 976 & 923 \\ 1339 & 768 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 623 & 832 \\ 572 & 207 \end{array}\right),\left(\begin{array}{rr} 768 & 1313 \\ 1027 & 534 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 38025 = 3^{2} \cdot 5^{2} \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 76050ew
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 390g4, its twist by $-195$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-65}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-390}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.53301680640000.120 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.6064546775040000.456 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | ss | add | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | - | - | 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 |
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.