Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-108528x+13715688\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-108528xz^2+13715688z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-140652963x+642028930398\)
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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 |
|---|---|---|
| $(203, 217)$ | $0.92811942884856927347571322756$ | $\infty$ |
| $(463, 7705)$ | $4.6480840941005330840257097226$ | $\infty$ |
| $(763/4, -763/8)$ | $0$ | $2$ |
Integral points
\( \left(193, -23\right) \), \( \left(193, -170\right) \), \( \left(203, 217\right) \), \( \left(203, -420\right) \), \( \left(281, 2206\right) \), \( \left(281, -2487\right) \), \( \left(463, 7705\right) \), \( \left(463, -8168\right) \), \( \left(2447, 118809\right) \), \( \left(2447, -121256\right) \)
Invariants
| Conductor: | $N$ | = | \( 51870 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $6345626621880$ | = | $2^{3} \cdot 3 \cdot 5 \cdot 7^{4} \cdot 13^{2} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{141369383441705190409}{6345626621880} \) | = | $2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{-2} \cdot 19^{-4} \cdot 473579^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5333298852063022813058570350$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5333298852063022813058570350$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9369267896600582$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.273746192729667$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.1758010389085777008718254089$ |
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| Real period: | $\Omega$ | ≈ | $0.70829116044328241260822195423$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot1\cdot1\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!$ | ≈ | $5.9153659272576415855999045386 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.915365927 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.708291 \cdot 4.175801 \cdot 8}{2^2} \\ & \approx 5.915365927\end{aligned}$$
Modular invariants
Modular form 51870.2.a.d
For more coefficients, see the Downloads section to the right.
| Modular degree: | 245760 |
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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 semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $2$ | $I_{4}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 15954 & 15955 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15953 & 8 \\ 15952 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 13681 & 8 \\ 6844 & 33 \end{array}\right),\left(\begin{array}{rr} 9976 & 2003 \\ 13969 & 13998 \end{array}\right),\left(\begin{array}{rr} 12776 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 5324 & 1 \\ 10663 & 6 \end{array}\right),\left(\begin{array}{rr} 4201 & 8 \\ 844 & 33 \end{array}\right),\left(\begin{array}{rr} 13968 & 5993 \\ 13993 & 14040 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$183000209817600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 8645 = 5 \cdot 7 \cdot 13 \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 10374 = 2 \cdot 3 \cdot 7 \cdot 13 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 7410 = 2 \cdot 3 \cdot 5 \cdot 13 \cdot 19 \) |
| $13$ | split multiplicative | $14$ | \( 3990 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 51870a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | nonsplit | nonsplit | nonsplit | nonsplit | ss | split | ord | nonsplit | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | 2 | 4 | 2,2 | 5 | 2 | 4 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.