Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-232416803x+1363776582848\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-232416803xz^2+1363776582848z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-301212176067x+63629263885896126\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9074, 39930)$ | $1.3011657150646845878496128928$ | $\infty$ |
| $(35207/4, -35211/8)$ | $0$ | $2$ |
Integral points
\( \left(9074, 39930\right) \), \( \left(9074, -49005\right) \)
Invariants
| Conductor: | $N$ | = | \( 25410 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $638027694150$ | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{4} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{783736670177727068275201}{360150} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-4} \cdot 92198401^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9933770532947483189840100364$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7944294168955630469530382474$ |
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| $abc$ quality: | $Q$ | ≈ | $1.074666741320478$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.8427910003154695$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3011657150646845878496128928$ |
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| Real period: | $\Omega$ | ≈ | $0.25984421006050586052987717704$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot1\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4096060382207394705747195760 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $4$ = $2^2$ (rounded) |
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BSD formula
$$\begin{aligned} 5.409606038 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.259844 \cdot 1.301166 \cdot 16}{2^2} \\ & \approx 5.409606038\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2621440 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_0^{*}$ | 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 | 16.48.0.122 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 36960 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 23 & 18 \\ 34398 & 34955 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 36929 & 32 \\ 36928 & 33 \end{array}\right),\left(\begin{array}{rr} 10110 & 23551 \\ 32329 & 16644 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 36959 \end{array}\right),\left(\begin{array}{rr} 22276 & 10109 \\ 671 & 31010 \end{array}\right),\left(\begin{array}{rr} 19966 & 10109 \\ 26081 & 31010 \end{array}\right),\left(\begin{array}{rr} 29591 & 26906 \\ 5478 & 34475 \end{array}\right),\left(\begin{array}{rr} 10583 & 26906 \\ 33462 & 3851 \end{array}\right)$.
The torsion field $K:=\Q(E[36960])$ is a degree-$313918488576000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/36960\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 363 = 3 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 25410.bl
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210.e1, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{11})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{11})\) | \(\Z/8\Z\) | 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 |
| $8$ | 8.8.77720518656.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | nonsplit | split | split | split | add | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 2 | 2 | 2 | - | 3 | 1 | 3 | 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 |
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.