Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-1143716x-470501899\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-1143716xz^2-470501899z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-18299451x-30130420970\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(15428985/64, 60542254439/512)$ | $12.511264050250200307489678729$ | $\infty$ |
| $(-2469/4, 2465/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 71478 \) | = | $2 \cdot 3^{2} \cdot 11 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $2263565518434$ | = | $2 \cdot 3^{7} \cdot 11 \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4824238966273}{66} \) | = | $2^{-1} \cdot 3^{-1} \cdot 11^{-1} \cdot 61^{3} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9265717330666042042158795254$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.094953900850670871486256809006$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0237597269657934$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.78324131704239$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.511264050250200307489678729$ |
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| Real period: | $\Omega$ | ≈ | $0.14598896698614592137520451130$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.3060260575477231943430281185 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.306026058 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.145989 \cdot 12.511264 \cdot 16}{2^2} \\ & \approx 7.306026058\end{aligned}$$
Modular invariants
Modular form 71478.2.a.bl
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5016 = 2^{3} \cdot 3 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3248 & 2109 \\ 3515 & 2374 \end{array}\right),\left(\begin{array}{rr} 4656 & 2945 \\ 4579 & 2718 \end{array}\right),\left(\begin{array}{rr} 2639 & 0 \\ 0 & 5015 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5009 & 8 \\ 5008 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4067 & 3268 \\ 4370 & 1293 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 5010 & 5011 \end{array}\right),\left(\begin{array}{rr} 1160 & 2907 \\ 3629 & 2642 \end{array}\right)$.
The torsion field $K:=\Q(E[5016])$ is a degree-$2496282624000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5016\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 35739 = 3^{2} \cdot 11 \cdot 19^{2} \) |
| $3$ | additive | $8$ | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 71478ct
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 66b3, its twist by $57$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-114}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-209}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{66}, \sqrt{-114})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | add | ord | ord | split | ord | ord | add | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | - | 1 | 1 | 2 | 3 | 1 | - | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.