Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+375x-12344\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+375xz^2-12344z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+6000x-790000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(100, 1012)$ | $0.12794045225631289494197893162$ | $\infty$ |
Integral points
\( \left(19, 40\right) \), \( \left(19, -41\right) \), \( \left(25, 112\right) \), \( \left(25, -113\right) \), \( \left(50, 362\right) \), \( \left(50, -363\right) \), \( \left(100, 1012\right) \), \( \left(100, -1013\right) \), \( \left(286, 4846\right) \), \( \left(286, -4847\right) \)
Invariants
| Conductor: | $N$ | = | \( 225 \) | = | $3^{2} \cdot 5^{2}$ |
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| Discriminant: | $\Delta$ | = | $-69198046875$ | = | $-1 \cdot 3^{11} \cdot 5^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{20480}{243} \) | = | $2^{12} \cdot 3^{-5} \cdot 5$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.76092676645643221831281035382$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.86133798616702287711865182013$ |
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| $abc$ quality: | $Q$ | ≈ | $1.131038420174359$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.975693472484786$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.12794045225631289494197893162$ |
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| Real period: | $\Omega$ | ≈ | $0.53919119789250514534088450402$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.82781238853188255079559880916 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.827812389 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.539191 \cdot 0.127940 \cdot 12}{1^2} \\ & \approx 0.827812389\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 240 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 30.48.1-30.d.1.1, level \( 30 = 2 \cdot 3 \cdot 5 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 20 \\ 5 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 29 & 20 \\ 0 & 23 \end{array}\right),\left(\begin{array}{rr} 21 & 10 \\ 20 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 5 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[30])$ is a degree-$2880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/30\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 25 = 5^{2} \) |
| $5$ | additive | $14$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 225e
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 75c1, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1350000.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.2.110716875.1 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.1822500000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.4.274968333542346954345703125.2 | \(\Z/5\Z\) | 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 | ss | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | ? | - | - | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | ? | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
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.