Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1822x+30393\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1822xz^2+30393z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-29155x+1915998\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(99/4, -99/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 49 \) | = | $7^{2}$ |
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| Discriminant: | $\Delta$ | = | $40353607$ | = | $7^{9} $ |
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| j-invariant: | $j$ | = | \( 16581375 \) | = | $3^{3} \cdot 5^{3} \cdot 17^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-7}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.52039027575200689013698072272$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.93904233603947808869203383486$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1980441775334598$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.771469225736265$ | |||
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: | $\Omega$ | ≈ | $1.9333117056168115467330768390$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.96665585280840577336653841951 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.966655853 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.933312 \cdot 1.000000 \cdot 2}{2^2} \\ & \approx 0.966655853\end{aligned}$$
Modular invariants
\( q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} - 3 q^{18} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 14 |
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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 is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 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 |
|---|---|---|
| $7$ | 7B.1.2 | 7.48.0.3 |
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 |
|---|---|---|---|
| $7$ | additive | $20$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 49.a
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49.a3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.28.1-49.1-a5 |
| $3$ | \(\Q(\zeta_{7})^+\) | \(\Z/14\Z\) | 3.3.49.1-49.1-a1 |
| $4$ | 4.0.1372.1 | \(\Z/4\Z\) | not in database |
| $6$ | \(\Q(\zeta_{28})^+\) | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | 8.0.30118144.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.7710244864.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.481890304.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.30118144.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.257298363.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.0.126548911552.1 | \(\Z/28\Z\) | not in database |
| $16$ | 16.0.59447875862838378496.4 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | 16.0.232218265089212416.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | 16.0.66202447602479769.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $20$ | 20.0.11194501700250570391613.1 | \(\Z/22\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | ord | ss | ss | add |
| $\lambda$-invariant(s) | ? | 0,0 | 0,0 | - |
| $\mu$-invariant(s) | ? | 0,0 | 0,0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry ? indicates that the invariants have not yet been computed.
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 $0$.