Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+1431x-46899\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+1431xz^2-46899z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+22893x-2978642\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(66, 543)$ | $1.0088998329368147078981668371$ | $\infty$ |
Integral points
\( \left(25, 51\right) \), \( \left(25, -76\right) \), \( \left(66, 543\right) \), \( \left(66, -609\right) \)
Invariants
| Conductor: | $N$ | = | \( 774 \) | = | $2 \cdot 3^{2} \cdot 43$ |
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| Discriminant: | $\Delta$ | = | $-1123219685376$ | = | $-1 \cdot 2^{14} \cdot 3^{13} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{444369620591}{1540767744} \) | = | $2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 43^{-1} \cdot 587^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.99527306746493756459326025751$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.44596692313088271889563763905$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9966384920336764$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.264821028210106$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0088998329368147078981668371$ |
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| Real period: | $\Omega$ | ≈ | $0.44310020709000863212531192490$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.7881748996295108356176826993 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.788174900 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.443100 \cdot 1.008900 \cdot 4}{1^2} \\ & \approx 1.788174900\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1344 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}$ | nonsplit multiplicative | 1 | 1 | 14 | 14 |
| $3$ | $2$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $43$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3612 = 2^{2} \cdot 3 \cdot 7 \cdot 43 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1807 & 14 \\ 1813 & 99 \end{array}\right),\left(\begin{array}{rr} 2407 & 3598 \\ 2401 & 3513 \end{array}\right),\left(\begin{array}{rr} 1807 & 14 \\ 0 & 1291 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3599 & 14 \\ 3598 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 3529 & 14 \\ 3031 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[3612])$ is a degree-$322962038784$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3612\Z)$.
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 | $4$ | \( 387 = 3^{2} \cdot 43 \) |
| $3$ | additive | $8$ | \( 86 = 2 \cdot 43 \) |
| $7$ | good | $2$ | \( 387 = 3^{2} \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 774.c
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 258.f2, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/7\Z\) | 2.0.3.1-22188.2-i2 |
| $3$ | 3.1.516.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.137388096.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.798768.1 | \(\Z/14\Z\) | not in database |
| $8$ | 8.2.9690085451952.4 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/21\Z\) | 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 | nonsplit | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | split | ord |
| $\lambda$-invariant(s) | 3 | - | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 | 2 | 1 |
| $\mu$-invariant(s) | 0 | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.