Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-56595x-395822\)
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(homogenize, simplify) |
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\(y^2z=x^3-56595xz^2-395822z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-56595x-395822\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(287, 2646)$ | $1.4055944608974614576316648862$ | $\infty$ |
| $(-7, 0)$ | $0$ | $2$ |
Integral points
\((-199,\pm 1728)\), \( \left(-7, 0\right) \), \((287,\pm 2646)\), \((722,\pm 18306)\)
Invariants
| Conductor: | $N$ | = | \( 134064 \) | = | $2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $11533816974016512$ | = | $2^{18} \cdot 3^{9} \cdot 7^{6} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{57066625}{32832} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3} \cdot 19^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7716043010377047130952360625$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44380409838395209457229504914$ |
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| $abc$ quality: | $Q$ | ≈ | $1.047658380732593$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.764556799613546$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4055944608974614576316648862$ |
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| Real period: | $\Omega$ | ≈ | $0.33670445339016313958821262630$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.5723186343155327525121140768 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.572318634 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.336704 \cdot 1.405594 \cdot 64}{2^2} \\ & \approx 7.572318634\end{aligned}$$
Modular invariants
Modular form 134064.2.a.dg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 663552 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1597 & 924 \\ 462 & 2353 \end{array}\right),\left(\begin{array}{rr} 2794 & 1827 \\ 1029 & 904 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 2806 & 1813 \\ 315 & 20 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3181 & 12 \\ 3180 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2279 & 0 \\ 0 & 3191 \end{array}\right),\left(\begin{array}{rr} 1497 & 2030 \\ 2506 & 307 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 3142 & 3183 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$190625218560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $2$ | \( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \) |
| $3$ | additive | $2$ | \( 14896 = 2^{4} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $26$ | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 134064bo
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 114a1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.715008.8 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-21}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.25747259328.5 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1661007193767936.116 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4601127960576.53 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
| $18$ | 18.0.63919367584391733682379810747781130223616.1 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ss | add | ss | ord | ord | split | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1,1 | - | 1,3 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 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.