Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-205562237x+419103349445\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-205562237xz^2+419103349445z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3288995787x+26819325368710\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8331/4, -8335/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 32634 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $480055865737598038302769152$ | = | $2^{14} \cdot 3^{18} \cdot 7^{9} \cdot 37^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{32654559320436510127}{16318562238480384} \) | = | $2^{-14} \cdot 3^{-12} \cdot 11^{3} \cdot 37^{-4} \cdot 277^{3} \cdot 1049^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8124934978929929189753449920$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8037547417674530944487078160$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0492480861640936$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.642610483665774$ | |||
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: | $\Omega$ | ≈ | $0.046490625535175081234273830070$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 224 $ = $ ( 2 \cdot 7 )\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6034750299698045491193344839 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.603475030 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.046491 \cdot 1.000000 \cdot 224}{2^2} \\ & \approx 2.603475030\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26492928 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $37$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 4.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6216 = 2^{3} \cdot 3 \cdot 7 \cdot 37 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 4668 & 1573 \\ 3101 & 1532 \end{array}\right),\left(\begin{array}{rr} 3113 & 1556 \\ 1576 & 3117 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 4448 & 3 \\ 1733 & 6200 \end{array}\right),\left(\begin{array}{rr} 6209 & 8 \\ 6208 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2185 & 8 \\ 2524 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4145 & 8 \\ 4148 & 33 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 6208 & 6195 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right)$.
The torsion field $K:=\Q(E[6216])$ is a degree-$5642506469376$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6216\Z)$.
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 | $4$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 3626 = 2 \cdot 7^{2} \cdot 37 \) |
| $7$ | additive | $20$ | \( 333 = 3^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 32634by
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 10878g2, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.12348.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.49392.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2439569664.6 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 37 |
|---|---|---|---|---|---|
| Reduction type | split | add | ord | add | nonsplit |
| $\lambda$-invariant(s) | 5 | - | 2 | - | 0 |
| $\mu$-invariant(s) | 0 | - | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ 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 $0$.