Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+1077x+6257\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+1077xz^2+6257z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1395765x+287739270\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(46, 369\right) \) | $0.24484897921462625601479905948$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([46:369:1]\) | $0.24484897921462625601479905948$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1659, 84672\right) \) | $0.24484897921462625601479905948$ | $\infty$ |
Integral points
\( \left(-2, 65\right) \), \( \left(-2, -63\right) \), \( \left(46, 369\right) \), \( \left(46, -415\right) \), \( \left(76, 689\right) \), \( \left(76, -765\right) \)
\([-2:65:1]\), \([-2:-63:1]\), \([46:369:1]\), \([46:-415:1]\), \([76:689:1]\), \([76:-765:1]\)
\((-69,\pm 13824)\), \((1659,\pm 84672)\), \((2739,\pm 157032)\)
Invariants
| Conductor: | $N$ | = | \( 2450 \) | = | $2 \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-96378060800$ | = | $-1 \cdot 2^{15} \cdot 5^{2} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{46969655}{32768} \) | = | $2^{-15} \cdot 5 \cdot 211^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.79616162271782996549647361019$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44503310388217674948966265040$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0629647337743247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.172219733999252$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.24484897921462625601479905948$ |
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| Real period: | $\Omega$ | ≈ | $0.67509797416375523430418968868$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 30 $ = $ ( 3 \cdot 5 )\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9589114953157279568968685782 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.958911495 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.675098 \cdot 0.244849 \cdot 30}{1^2} \\ & \approx 4.958911495\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2160 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $5$ | 5B.4.1 | 5.12.0.1 | $12$ |
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 $384$, genus $9$, and generators
$\left(\begin{array}{rr} 631 & 210 \\ 735 & 211 \end{array}\right),\left(\begin{array}{rr} 721 & 120 \\ 720 & 721 \end{array}\right),\left(\begin{array}{rr} 281 & 210 \\ 280 & 1 \end{array}\right),\left(\begin{array}{rr} 421 & 210 \\ 525 & 211 \end{array}\right),\left(\begin{array}{rr} 1 & 672 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 120 & 1 \end{array}\right),\left(\begin{array}{rr} 599 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 281 & 560 \\ 280 & 561 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 630 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 770 \\ 280 & 337 \end{array}\right),\left(\begin{array}{rr} 1 & 756 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 526 & 105 \\ 525 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 42 \\ 210 & 421 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | good | $2$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $10$ | \( 49 = 7^{2} \) |
| $7$ | additive | $26$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 2450v
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50b2, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/5\Z\) | 2.0.7.1-2500.2-b4 |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-15})\) | \(\Z/15\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.781396875.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.1852200000.5 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.13720000.1 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.12047257600000000.3 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/30\Z\) | not in database |
| $18$ | 18.6.38465045506169224763625000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.125070513254830584000000000000000.1 | \(\Z/6\Z\) | not in database |
| $20$ | 20.4.1315377880819141864776611328125.1 | \(\Z/5\Z\) | 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 | split | ord | add | add | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 5 | - | - | 1 | 1 | 3 | 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 |
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.