Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+49x-686\)
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(homogenize, simplify) |
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\(y^2z=x^3+49xz^2-686z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+49x-686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9, 22\right) \) | $1.8919590029013207883469938310$ | $\infty$ |
| \( \left(21, 98\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([9:22:1]\) | $1.8919590029013207883469938310$ | $\infty$ |
| \([21:98:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(9, 22\right) \) | $1.8919590029013207883469938310$ | $\infty$ |
| \( \left(21, 98\right) \) | $0$ | $4$ |
Integral points
\( \left(7, 0\right) \), \((9,\pm 22)\), \((21,\pm 98)\), \((70,\pm 588)\), \((105,\pm 1078)\)
\([7:0:1]\), \([9:\pm 22:1]\), \([21:\pm 98:1]\), \([70:\pm 588:1]\), \([105:\pm 1078:1]\)
\( \left(7, 0\right) \), \((9,\pm 22)\), \((21,\pm 98)\), \((70,\pm 588)\), \((105,\pm 1078)\)
Invariants
| Conductor: | $N$ | = | \( 392 \) | = | $2^{3} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-210827008$ | = | $-1 \cdot 2^{8} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{432}{7} \) | = | $2^{4} \cdot 3^{3} \cdot 7^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.27777506716912385478890071327$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1572781277318296707085970728$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8915192755066496$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.45214118997548$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8919590029013207883469938310$ |
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| Real period: | $\Omega$ | ≈ | $0.86788635344346877622337901890$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6420053998925684615630104991 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.642005400 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.867886 \cdot 1.891959 \cdot 16}{4^2} \\ & \approx 1.642005400\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 96 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.24.0.51 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.48.0-56.bn.1.4, level \( 56 = 2^{3} \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 43 & 38 \\ 26 & 51 \end{array}\right),\left(\begin{array}{rr} 32 & 53 \\ 27 & 54 \end{array}\right),\left(\begin{array}{rr} 25 & 24 \\ 6 & 31 \end{array}\right),\left(\begin{array}{rr} 49 & 8 \\ 48 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 50 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$64512$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $32$ | \( 8 = 2^{3} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 392.d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 56.a4, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.7.1-448.4-a1 |
| $4$ | \(\Q(\sqrt{14 +4 \sqrt{14}})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.30118144.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.157351936.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.7710244864.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.263473523712.3 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | ss | ord | add | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3,1 | 1 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 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.