Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-601055561x+5671691591691\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-601055561xz^2+5671691591691z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-778968007083x+264620779805956518\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(57619/4, 352511/8)$ | $0.91058040205133160593385611110$ | $\infty$ |
| $(56751/4, -56751/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 45570 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $230419841138488059537150$ | = | $2 \cdot 3^{8} \cdot 5^{2} \cdot 7^{7} \cdot 31^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{204117072508351537504018561}{1958536333827640350} \) | = | $2^{-1} \cdot 3^{-8} \cdot 5^{-2} \cdot 7^{-1} \cdot 23^{3} \cdot 31^{-8} \cdot 1103^{3} \cdot 23209^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6445533529120620575239706231$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.6715982783844054049712942514$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0222003230459995$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.735915348216759$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.91058040205133160593385611110$ |
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| Real period: | $\Omega$ | ≈ | $0.089592162878772267397617585881$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 512 $ = $ 1\cdot2^{3}\cdot2\cdot2^{2}\cdot2^{3} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.442351064934507278467725430 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.442351065 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.089592 \cdot 0.910580 \cdot 512}{2^2} \\ & \approx 10.442351065\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15728640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $31$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.48.0.211 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3472 = 2^{4} \cdot 7 \cdot 31 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 3457 & 16 \\ 3456 & 17 \end{array}\right),\left(\begin{array}{rr} 561 & 16 \\ 1016 & 129 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 884 & 869 \\ 1895 & 1746 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3374 & 3459 \end{array}\right),\left(\begin{array}{rr} 1968 & 3467 \\ 541 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1310 & 869 \\ 2249 & 1746 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 3468 & 3469 \end{array}\right)$.
The torsion field $K:=\Q(E[3472])$ is a degree-$230385254400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3472\Z)$.
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 | $4$ | \( 49 = 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 15190 = 2 \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 9114 = 2 \cdot 3 \cdot 7^{2} \cdot 31 \) |
| $7$ | additive | $32$ | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 45570da
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 6510r5, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.308409794560000.68 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.308409794560000.70 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | nonsplit | add | ord | ord | ord | ord | ss | ord | split | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 12 | 2 | 1 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 2 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | 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.