Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+269505x+7364803\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+269505xz^2+7364803z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4312077x+475659470\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-109/4, 109/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 67518 \) | = | $2 \cdot 3^{2} \cdot 11^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-1276656171939321822$ | = | $-1 \cdot 2 \cdot 3^{8} \cdot 11^{12} \cdot 31 $ |
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| j-invariant: | $j$ | = | \( \frac{1676253304439}{988531038} \) | = | $2^{-1} \cdot 3^{-2} \cdot 7^{3} \cdot 11^{-6} \cdot 31^{-1} \cdot 1697^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1636008685842738292947029784$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41534708785103371156610857096$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9679342602005597$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.417802798102853$ | |||
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: | $\Omega$ | ≈ | $0.16547772545804308630445758062$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\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)$ | ≈ | $2.6476436073286893808713212899 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.647643607 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.165478 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 2.647643607\end{aligned}$$
Modular invariants
Modular form 67518.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2211840 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $31$ | $1$ | $I_{1}$ | 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 \( 8184 = 2^{3} \cdot 3 \cdot 11 \cdot 31 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 2114 & 1 \\ 7127 & 0 \end{array}\right),\left(\begin{array}{rr} 8181 & 4 \\ 8180 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1025 & 7162 \\ 7160 & 1023 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 4091 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 5953 & 4 \\ 3722 & 9 \end{array}\right),\left(\begin{array}{rr} 2729 & 4 \\ 5458 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[8184])$ is a degree-$72406794240000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8184\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 33759 = 3^{2} \cdot 11^{2} \cdot 31 \) |
| $3$ | additive | $8$ | \( 7502 = 2 \cdot 11^{2} \cdot 31 \) |
| $11$ | additive | $72$ | \( 558 = 2 \cdot 3^{2} \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 67518ba
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2046d2, its twist by $33$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-62}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.1080288.1 | \(\Z/4\Z\) | 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 | 11 | 31 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | add | split |
| $\lambda$-invariant(s) | 10 | - | - | 1 |
| $\mu$-invariant(s) | 1 | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.