Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-570407x-165561249\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-570407xz^2-165561249z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9126507x-10605046426\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3455, 196002)$ | $2.0059447291072489909189333160$ | $\infty$ |
| $(-445, 222)$ | $0$ | $2$ |
Integral points
\( \left(-445, 222\right) \), \( \left(3455, 196002\right) \), \( \left(3455, -199458\right) \)
Invariants
| Conductor: | $N$ | = | \( 106470 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $15984667657969920$ | = | $2^{8} \cdot 3^{7} \cdot 5 \cdot 7 \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{5832972054001}{4542720} \) | = | $2^{-8} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{-2} \cdot 47^{3} \cdot 383^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0404988989584548956949314628$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.20871807589363168197056512356$ |
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| $abc$ quality: | $Q$ | ≈ | $0.907047640144071$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.438287033835674$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.0059447291072489909189333160$ |
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| Real period: | $\Omega$ | ≈ | $0.17372955461335527372262502864$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{3}\cdot2^{2}\cdot1\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.151740299737918840106558514 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.151740300 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.173730 \cdot 2.005945 \cdot 128}{2^2} \\ & \approx 11.151740300\end{aligned}$$
Modular invariants
Modular form 106470.2.a.fo
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1376256 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 7264 & 21835 \\ 45 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 21742 & 21827 \end{array}\right),\left(\begin{array}{rr} 15616 & 5 \\ 9315 & 21826 \end{array}\right),\left(\begin{array}{rr} 21827 & 21824 \\ 5296 & 315 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 5460 & 5461 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 19121 & 16 \\ 13586 & 16287 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 21836 & 21837 \end{array}\right),\left(\begin{array}{rr} 4376 & 1 \\ 13183 & 10 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 | $4$ | \( 53235 = 3^{2} \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $3$ | additive | $8$ | \( 11830 = 2 \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 21294 = 2 \cdot 3^{2} \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 15210 = 2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 106470gb
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2730w1, its twist by $-39$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-455}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-39}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-39})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-39})\) | \(\Z/8\Z\) | 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 |
| $8$ | 8.0.3471607400625.5 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | split | split | ord | add | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 5 | - | 2 | 2 | 1 | - | 3 | 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 |
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.