Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-965x-13940\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-965xz^2-13940z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15435x-907578\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(46, 170\right) \) | $3.0685237365405596481302553212$ | $\infty$ |
| \( \left(37, -19\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([46:170:1]\) | $3.0685237365405596481302553212$ | $\infty$ |
| \([37:-19:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(183, 1548\right) \) | $3.0685237365405596481302553212$ | $\infty$ |
| \( \left(147, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(37, -19\right) \), \( \left(46, 170\right) \), \( \left(46, -217\right) \), \( \left(380, 7184\right) \), \( \left(380, -7565\right) \)
\([37:-19:1]\), \([46:170:1]\), \([46:-217:1]\), \([380:7184:1]\), \([380:-7565:1]\)
\( \left(147, 0\right) \), \((183,\pm 1548)\), \((1519,\pm 58996)\)
Invariants
| Conductor: | $N$ | = | \( 441 \) | = | $3^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-29417779503$ | = | $-1 \cdot 3^{6} \cdot 7^{9} $ |
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| j-invariant: | $j$ | = | \( -3375 \) | = | $-1 \cdot 3^{3} \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.72312282980608908112598728046$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2856159263194507434006498956$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9802957926219806$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.360848806714531$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0685237365405596481302553212$ |
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| Real period: | $\Omega$ | ≈ | $0.42188320156824859739386023495$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.2945586180598962800435659133 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.294558618 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.421883 \cdot 3.068524 \cdot 4}{2^2} \\ & \approx 1.294558618\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 224 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 49 = 7^{2} \) |
| $7$ | additive | $20$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 7 and 14.
Its isogeny class 441.c
consists of 4 curves linked by isogenies of
degrees dividing 14.
Twists
The minimal quadratic twist of this elliptic curve is 49.a4, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.7.1-3969.1-CMa1 |
| $4$ | \(\Q(\sqrt[4]{252})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{42 +18 \sqrt{-7}})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.0.64827.1 | \(\Z/14\Z\) | not in database |
| $8$ | 8.0.2439569664.6 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2439569664.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.257298363.1 | \(\Z/6\Z\) | not in database |
| $12$ | \(\Q(\zeta_{21})\) | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | 16.0.66202447602479769.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $20$ | 20.0.661024130898095931054356037.1 | \(\Z/2\Z \oplus \Z/22\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | add | ss | add | ord | ss | ss | ss | ord | ord | ss | ord | ss | ord | ss |
| $\lambda$-invariant(s) | ? | - | 1,1 | - | 1 | 1,1 | 1,1 | 1,1 | 1 | 1 | 3,1 | 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 |
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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.