Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-71018787x+230360484130\)
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(homogenize, simplify) |
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\(y^2z=x^3-71018787xz^2+230360484130z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-71018787x+230360484130\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4767, 11858)$ | $0.71614495476563823986886403192$ | $\infty$ |
| $(4865, 0)$ | $0$ | $2$ |
Integral points
\((-9247,\pm 310464)\), \((4767,\pm 11858)\), \( \left(4865, 0\right) \), \((4866,\pm 22)\), \((5009,\pm 17424)\)
Invariants
| Conductor: | $N$ | = | \( 77616 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $678922791477460992$ | = | $2^{14} \cdot 3^{7} \cdot 7^{6} \cdot 11^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{112763292123580561}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{-5} \cdot 179^{3} \cdot 2699^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9632616658182044657828067657$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.74785326639654765811527565406$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0637869122465706$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.848288041806828$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.71614495476563823986886403192$ |
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| Real period: | $\Omega$ | ≈ | $0.20518583169750534683613483074$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2^{2}\cdot2\cdot2\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9388559631911965510324957473 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.938855963 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.205186 \cdot 0.716145 \cdot 80}{2^2} \\ & \approx 2.938855963\end{aligned}$$
Modular invariants
Modular form 77616.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6912000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 | 8.6.0.4 |
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 6929 & 6580 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 5704 & 3955 \\ 1365 & 6614 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 9000 & 8891 \end{array}\right),\left(\begin{array}{rr} 9221 & 20 \\ 9220 & 21 \end{array}\right),\left(\begin{array}{rr} 4621 & 2660 \\ 1330 & 8121 \end{array}\right),\left(\begin{array}{rr} 3959 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 736 & 5285 \\ 5355 & 2626 \end{array}\right),\left(\begin{array}{rr} 3697 & 2660 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$3269984256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $2$ | \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| $5$ | good | $2$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 77616.i
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 66.c1, its twist by $-84$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.413952.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.4.882000.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.0.186606965293056.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.793808535953664.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.8.11389585284000000.13 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.0.52929948960000000000.2 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.72665756964764081922245052441600000000000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | 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 | ord | add | split | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | - | 4 | 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 |
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.