Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-859140x+306223300\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-859140xz^2+306223300z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-69590367x+223445556774\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(535, 0)$ | $0$ | $2$ |
Integral points
\( \left(535, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 2420 \) | = | $2^{2} \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $15090865222400$ | = | $2^{8} \cdot 5^{2} \cdot 11^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{154639330142416}{33275} \) | = | $2^{4} \cdot 5^{-2} \cdot 7^{3} \cdot 11^{-3} \cdot 17^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9133979224045684873611741537$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.25235216563208634238538095041$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9734387724171747$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.751525590231188$ | |||
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: | $\Omega$ | ≈ | $0.55554614620751354101906640851$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 3\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6666384386225406230571992255 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.666638439 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.555546 \cdot 1.000000 \cdot 12}{2^2} \\ & \approx 1.666638439\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 25920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 606 & 397 \\ 605 & 386 \end{array}\right),\left(\begin{array}{rr} 397 & 12 \\ 402 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 649 & 12 \\ 648 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 221 & 12 \\ 220 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 610 & 651 \end{array}\right),\left(\begin{array}{rr} 530 & 657 \\ 87 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[660])$ is a degree-$304128000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/660\Z)$.
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 | $2$ | \( 121 = 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 484 = 2^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 20 = 2^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 2420g
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 220a4, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.4400.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.119790000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.11610102169600.25 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.37480960000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.189747360000.2 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.50966421015780060734456530459200000000.2 | \(\Z/18\Z\) | 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 | 11 |
|---|---|---|---|---|
| Reduction type | add | ord | split | add |
| $\lambda$-invariant(s) | - | 4 | 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 $0$.