Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+1624x+4348\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+1624xz^2+4348z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2105325x+196555950\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1816/25, 84399/125)$ | $7.7965028795528156319540142913$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 15950 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29$ |
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| Discriminant: | $\Delta$ | = | $-282028298750$ | = | $-1 \cdot 2 \cdot 5^{4} \cdot 11 \cdot 29^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{758553353975}{451245278} \) | = | $2^{-1} \cdot 5^{2} \cdot 11^{-1} \cdot 29^{-5} \cdot 3119^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.88643094689599908544496791066$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.34995164275129896057804813292$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9409622865190032$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4919585581710537$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.7965028795528156319540142913$ |
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| Real period: | $\Omega$ | ≈ | $0.59572463968820147840198993487$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.6445688687496263818529814100 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.644568869 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.595725 \cdot 7.796503 \cdot 1}{1^2} \\ & \approx 4.644568869\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13200 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.1.3 | 5.24.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12760 = 2^{3} \cdot 5 \cdot 11 \cdot 29 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 9573 & 6390 \\ 20 & 12189 \end{array}\right),\left(\begin{array}{rr} 3191 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 12705 & 12641 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 6375 & 12756 \end{array}\right),\left(\begin{array}{rr} 11606 & 5 \\ 11595 & 12756 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3086 & 5 \\ 12315 & 12756 \end{array}\right),\left(\begin{array}{rr} 12751 & 10 \\ 12750 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[12760])$ is a degree-$138293084160000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12760\Z)$.
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 | $4$ | \( 7975 = 5^{2} \cdot 11 \cdot 29 \) |
| $5$ | additive | $14$ | \( 22 = 2 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 1450 = 2 \cdot 5^{2} \cdot 29 \) |
| $29$ | nonsplit multiplicative | $30$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 15950.i
consists of 2 curves linked by isogenies of
degree 5.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.63800.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{5})\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.10387762880000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.2.2679486012500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.0.35898226455915750781250000000000000000.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | ord | add | ord | split | ord | ord | ss | ord | nonsplit | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 3 | - | 3 | 2 | 1 | 1 | 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 |
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.