Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-25886720x-50688355293\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-25886720xz^2-50688355293z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-414187515x-3244468926250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-11749/4, 11745/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 35550 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $1092096000$ | = | $2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{568435245123699267862527}{323584} \) | = | $2^{-12} \cdot 3^{6} \cdot 7^{3} \cdot 79^{-1} \cdot 307^{3} \cdot 4283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4472185094618682405728893073$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7702059591863157240738881648$ |
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| $abc$ quality: | $Q$ | ≈ | $1.130124799199383$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.995147520641889$ | |||
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: | $\Omega$ | ≈ | $0.066931460138371905318247275617$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.2127100866418514552758692296 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.212710087 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.066931 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 3.212710087\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 946176 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $79$ | $1$ | $I_{1}$ | 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4740 = 2^{2} \cdot 3 \cdot 5 \cdot 79 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2848 & 1 \\ 2843 & 0 \end{array}\right),\left(\begin{array}{rr} 3482 & 1 \\ 2999 & 0 \end{array}\right),\left(\begin{array}{rr} 3164 & 1 \\ 1579 & 0 \end{array}\right),\left(\begin{array}{rr} 3557 & 1186 \\ 1184 & 3555 \end{array}\right),\left(\begin{array}{rr} 4737 & 4 \\ 4736 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[4740])$ is a degree-$7087266201600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4740\Z)$.
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 | $4$ | \( 1185 = 3 \cdot 5 \cdot 79 \) |
| $3$ | additive | $6$ | \( 1975 = 5^{2} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 35550.bc
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 35550.a1, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{1185}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.4266000.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
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 | 79 |
|---|---|---|---|---|
| Reduction type | split | add | add | split |
| $\lambda$-invariant(s) | 6 | - | - | 1 |
| $\mu$-invariant(s) | 0 | - | - | 0 |
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 $0$.