Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-1056x+13552\)
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(homogenize, simplify) |
\(y^2z=x^3-1056xz^2+13552z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-1056x+13552\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(33, 121)$ | $2.2534409880008662547433239938$ | $\infty$ |
Integral points
\((33,\pm 121)\)
Invariants
Conductor: | $N$ | = | \( 17424 \) | = | $2^{4} \cdot 3^{2} \cdot 11^{2}$ |
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Discriminant: | $\Delta$ | = | $-3974344704$ | = | $-1 \cdot 2^{12} \cdot 3^{6} \cdot 11^{3} $ |
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j-invariant: | $j$ | = | \( -32768 \) | = | $-1 \cdot 2^{15}$ |
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-11})/2]\) (potential complex multiplication) |
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.61938050056452570544368337978$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2225466425290670856866572546$ |
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$abc$ quality: | $Q$ | ≈ | $1.0251241218312794$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.333303796414238$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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Mordell-Weil rank: | $r$ | = | $ 1$ |
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.2534409880008662547433239938$ |
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Real period: | $\Omega$ | ≈ | $1.3863396215093464149038245325$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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Special value: | $ L'(E,1)$ | ≈ | $6.2480690527975371194162454824 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.248069053 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.386340 \cdot 2.253441 \cdot 2}{1^2} \\ & \approx 6.248069053\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 8640 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
$3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
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 | $2$ | \( 99 = 3^{2} \cdot 11 \) |
$3$ | additive | $2$ | \( 1936 = 2^{4} \cdot 11^{2} \) |
$11$ | additive | $42$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 17424.cb
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121.b2, its twist by $12$.
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 |
---|---|---|---|
$3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
$4$ | 4.0.63888.1 | \(\Z/3\Z\) | not in database |
$4$ | 4.2.191664.1 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$8$ | 8.0.36735088896.3 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$8$ | 8.0.4591886112000.13 | \(\Z/5\Z\) | not in database |
$10$ | 10.10.53339349076992.1 | \(\Z/11\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$12$ | deg 12 | \(\Z/9\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/5\Z\) | not in database |
$16$ | deg 16 | \(\Z/15\Z\) | 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 |
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Reduction type | add | add | ord | ss | add | ss | ss | ss | ord | ss | ord | ord | ss | ss | ord |
$\lambda$-invariant(s) | - | - | 1 | 1,5 | - | 1,3 | 1,1 | 1,1 | 1 | 1,1 | 1 | 1 | 1,1 | 1,3 | 1 |
$\mu$-invariant(s) | - | - | 0 | 0,0 | - | 0,0 | 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.