Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-12615x+536558\)
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(homogenize, simplify) |
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\(y^2z=x^3-12615xz^2+536558z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12615x+536558\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(58, 0)$ | $0$ | $2$ |
Integral points
\( \left(58, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 30276 \) | = | $2^{2} \cdot 3^{2} \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $4111418794752$ | = | $2^{8} \cdot 3^{3} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( 54000 \) | = | $2^{4} \cdot 3^{3} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-3}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2148062091988476126316048974$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2055928983347136967536638423$ |
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| $abc$ quality: | $Q$ | ≈ | $1.027195810121916$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.871011871304256$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.78113604070266234975328770434$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.78113604070266234975328770434 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.781136041 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.781136 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 0.781136041\end{aligned}$$
Modular invariants
\( q - 4 q^{7} + 2 q^{13} - 8 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 50400 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 2523 = 3 \cdot 29^{2} \) |
| $3$ | additive | $6$ | \( 3364 = 2^{2} \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 36 = 2^{2} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 30276.i
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 36.a2, its twist by $29$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{29}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.363312.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{29})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.853419888.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.135163503968256.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2111929749504.28 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.131995609344.1 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/14\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.12234312533745663808217190322176.1 | \(\Z/18\Z\) | 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 | 29 |
|---|---|---|---|
| Reduction type | add | add | add |
| $\lambda$-invariant(s) | - | - | - |
| $\mu$-invariant(s) | - | - | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.