Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-49x\)
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(homogenize, simplify) |
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\(y^2z=x^3-49xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-49x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(25, 120)$ | $2.9888815076033406098269149411$ | $\infty$ |
| $(0, 0)$ | $0$ | $2$ |
| $(7, 0)$ | $0$ | $2$ |
Integral points
\( \left(-7, 0\right) \), \( \left(0, 0\right) \), \( \left(7, 0\right) \), \((25,\pm 120)\)
Invariants
| Conductor: | $N$ | = | \( 1568 \) | = | $2^{5} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $7529536$ | = | $2^{6} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-1}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.0089957388961197890090173491465$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3105329259115095182522750833$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.1653259983490156$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9888815076033406098269149411$ |
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| Real period: | $\Omega$ | ≈ | $1.9820892034157685460763917976$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9621148832548133741401093524 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.962114883 \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 1.982089 \cdot 2.988882 \cdot 8}{4^2} \\ & \approx 2.962114883\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 5 | 6 | 0 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $2$ | 2Cs | 16.192.9.143 |
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 | $26$ | \( 32 = 2^{5} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1568g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 32a2, its twist by $28$.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $49$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.344128684032.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.19668992000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.101415451701035401216.11 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | 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 | ss | ord | ord | ss | ss | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | 1,1 | 1 | - | 1,1 | 1 | 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 | 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.