Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-13275x-465750\)
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(homogenize, simplify) |
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\(y^2z=x^3-13275xz^2-465750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13275x-465750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-41, 98)$ | $1.8887190127728822977936293808$ | $\infty$ |
| $(-65, 350)$ | $2.2899765659041331444365588469$ | $\infty$ |
| $(-90, 0)$ | $0$ | $2$ |
Integral points
\( \left(-90, 0\right) \), \((-65,\pm 350)\), \((-54,\pm 306)\), \((-41,\pm 98)\), \((135,\pm 450)\), \((351,\pm 6174)\), \((1846,\pm 79156)\)
Invariants
| Conductor: | $N$ | = | \( 25200 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $56010528000000$ | = | $2^{11} \cdot 3^{6} \cdot 5^{6} \cdot 7^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{11090466}{2401} \) | = | $2 \cdot 3^{3} \cdot 7^{-4} \cdot 59^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3519922737525175446514587481$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63741774231187068864567298164$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1170620055320515$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9562014727010815$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.1977970707818631321951253233$ |
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| Real period: | $\Omega$ | ≈ | $0.45161480750735977833655022658$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.5831492643044393955216917189 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.583149264 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.451615 \cdot 4.197797 \cdot 16}{2^2} \\ & \approx 7.583149264\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 65536 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 8.12.0.15 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 559 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 629 & 300 \\ 30 & 659 \end{array}\right),\left(\begin{array}{rr} 833 & 8 \\ 832 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 136 & 225 \\ 195 & 406 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 834 & 835 \end{array}\right),\left(\begin{array}{rr} 241 & 120 \\ 180 & 481 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$1486356480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \) |
| $5$ | additive | $14$ | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 25200.l
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 56.a2, its twist by $60$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.849346560000.29 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.124467840000.9 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2039281090560000.207 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.13442526720000.25 | \(\Z/6\Z\) | 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 | add | add | add | nonsplit | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | - | - | 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.