Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-530x+21852\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-530xz^2+21852z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-686907x+1021587606\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-26, 148)$ | $0.20848682207591385510173356389$ | $\infty$ |
Integral points
\( \left(-26, 148\right) \), \( \left(-26, -122\right) \), \( \left(18, 126\right) \), \( \left(18, -144\right) \), \( \left(82, 688\right) \), \( \left(82, -770\right) \)
Invariants
| Conductor: | $N$ | = | \( 55770 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-197589763800$ | = | $-1 \cdot 2^{3} \cdot 3^{12} \cdot 5^{2} \cdot 11 \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{97435188409}{1169170200} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-12} \cdot 5^{-2} \cdot 11^{-1} \cdot 13 \cdot 19^{3} \cdot 103^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.84929569923410317599700478723$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.42180413965718038665475688030$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9418422562332995$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.066273751671904$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.20848682207591385510173356389$ |
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| Real period: | $\Omega$ | ≈ | $0.85398658212241050705850334062$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 3\cdot( 2^{2} \cdot 3 )\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.819236299356441258007292992 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.819236299 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.853987 \cdot 0.208487 \cdot 72}{1^2} \\ & \approx 12.819236299\end{aligned}$$
Modular invariants
Modular form 55770.2.a.cy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 82944 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2718 & 721 \\ 3289 & 3147 \end{array}\right),\left(\begin{array}{rr} 1717 & 6 \\ 1719 & 19 \end{array}\right),\left(\begin{array}{rr} 937 & 6 \\ 2811 & 19 \end{array}\right),\left(\begin{array}{rr} 2575 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3427 & 6 \\ 3426 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1059 & 2 \\ 2386 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$1594117324800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $4$ | \( 1859 = 11 \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 9295 = 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 11154 = 2 \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $38$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 55770.cy
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.14872.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.19463521792.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.91734201219375.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.2875292992.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.882448648173776438820714689670148879348828125.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.24486073189633358000334391968528030144000000000000.1 | \(\Z/6\Z\) | 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 | split | ord | split | add | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 3 | 4 | 4 | 1 | 2 | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.