Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-3033x+65437\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-3033xz^2+65437z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-245700x+46966500\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 50\right) \) | $0.18645113743696470179906539771$ | $\infty$ |
| \( \left(12, 175\right) \) | $0.63226000006609433691579869812$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37:50:1]\) | $0.18645113743696470179906539771$ | $\infty$ |
| \([12:175:1]\) | $0.63226000006609433691579869812$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(330, 1350\right) \) | $0.18645113743696470179906539771$ | $\infty$ |
| \( \left(105, 4725\right) \) | $0.63226000006609433691579869812$ | $\infty$ |
Integral points
\((-63,\pm 50)\), \((-29,\pm 358)\), \((-23,\pm 350)\), \((12,\pm 175)\), \((27,\pm 50)\), \((28,\pm 41)\), \((33,\pm 14)\), \((37,\pm 50)\), \((51,\pm 202)\), \((61,\pm 322)\), \((187,\pm 2450)\), \((237,\pm 3550)\), \((8227,\pm 746150)\), \((13977,\pm 1652350)\)
\([-63:\pm 50:1]\), \([-29:\pm 358:1]\), \([-23:\pm 350:1]\), \([12:\pm 175:1]\), \([27:\pm 50:1]\), \([28:\pm 41:1]\), \([33:\pm 14:1]\), \([37:\pm 50:1]\), \([51:\pm 202:1]\), \([61:\pm 322:1]\), \([187:\pm 2450:1]\), \([237:\pm 3550:1]\), \([8227:\pm 746150:1]\), \([13977:\pm 1652350:1]\)
\((-63,\pm 50)\), \((-29,\pm 358)\), \((-23,\pm 350)\), \((12,\pm 175)\), \((27,\pm 50)\), \((28,\pm 41)\), \((33,\pm 14)\), \((37,\pm 50)\), \((51,\pm 202)\), \((61,\pm 322)\), \((187,\pm 2450)\), \((237,\pm 3550)\), \((8227,\pm 746150)\), \((13977,\pm 1652350)\)
Invariants
| Conductor: | $N$ | = | \( 9800 \) | = | $2^{3} \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-6860000000$ | = | $-1 \cdot 2^{8} \cdot 5^{7} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{2249728}{5} \) | = | $-1 \cdot 2^{10} \cdot 5^{-1} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.77016224761555302997584433257$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.98313236623862235654569493421$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8600818321634119$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8813007654845126$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.11654913681564331218284908928$ |
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| Real period: | $\Omega$ | ≈ | $1.3327352173970012936787795551$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.9705324541257355759860349912 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.970532454 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.332735 \cdot 0.116549 \cdot 32}{1^2} \\ & \approx 4.970532454\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9216 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | 1 | 3 | 8 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 70.2.0.a.1, level \( 70 = 2 \cdot 5 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 69 & 2 \\ 68 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 2 \\ 31 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 2 \\ 57 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 69 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[70])$ is a degree-$2903040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/70\Z)$.
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 | $2$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 392 = 2^{3} \cdot 7^{2} \) |
| $7$ | additive | $20$ | \( 200 = 2^{3} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 9800.m consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1960.h1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.140.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.686000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 4 | - | - | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.